MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  yonedainv Structured version   Visualization version   GIF version

Theorem yonedainv 18326
Description: The Yoneda Lemma with explicit inverse. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y 𝑌 = (Yon‘𝐶)
yoneda.b 𝐵 = (Base‘𝐶)
yoneda.1 1 = (Id‘𝐶)
yoneda.o 𝑂 = (oppCat‘𝐶)
yoneda.s 𝑆 = (SetCat‘𝑈)
yoneda.t 𝑇 = (SetCat‘𝑉)
yoneda.q 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h 𝐻 = (HomF𝑄)
yoneda.r 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (𝜑𝐶 ∈ Cat)
yoneda.w (𝜑𝑉𝑊)
yoneda.u (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
yoneda.v (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
yoneda.m 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
yonedainv.i 𝐼 = (Inv‘𝑅)
yonedainv.n 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
Assertion
Ref Expression
yonedainv (𝜑𝑀(𝑍𝐼𝐸)𝑁)
Distinct variable groups:   𝑓,𝑎,𝑔,𝑥,𝑦, 1   𝑢,𝑎,𝑔,𝑦,𝐶,𝑓,𝑥   𝐸,𝑎,𝑓,𝑔,𝑢,𝑦   𝐵,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑁,𝑎   𝑂,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑆,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑔,𝑀,𝑢,𝑦   𝑄,𝑎,𝑓,𝑔,𝑢,𝑥   𝑇,𝑓,𝑔,𝑢,𝑦   𝜑,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑢,𝑅   𝑌,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑍,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦
Allowed substitution hints:   𝑄(𝑦)   𝑅(𝑥,𝑦,𝑓,𝑔,𝑎)   𝑇(𝑥,𝑎)   𝑈(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   1 (𝑢)   𝐸(𝑥)   𝐻(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝐼(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝑀(𝑥,𝑓,𝑎)   𝑁(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝑊(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)

Proof of Theorem yonedainv
Dummy variables 𝑏 𝑘 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 yoneda.r . . 3 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
2 eqid 2737 . . . 4 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
3 yoneda.q . . . . 5 𝑄 = (𝑂 FuncCat 𝑆)
43fucbas 18008 . . . 4 (𝑂 Func 𝑆) = (Base‘𝑄)
5 yoneda.o . . . . 5 𝑂 = (oppCat‘𝐶)
6 yoneda.b . . . . 5 𝐵 = (Base‘𝐶)
75, 6oppcbas 17761 . . . 4 𝐵 = (Base‘𝑂)
82, 4, 7xpcbas 18223 . . 3 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
9 eqid 2737 . . 3 ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
10 yoneda.y . . . . 5 𝑌 = (Yon‘𝐶)
11 yoneda.1 . . . . 5 1 = (Id‘𝐶)
12 yoneda.s . . . . 5 𝑆 = (SetCat‘𝑈)
13 yoneda.t . . . . 5 𝑇 = (SetCat‘𝑉)
14 yoneda.h . . . . 5 𝐻 = (HomF𝑄)
15 yoneda.e . . . . 5 𝐸 = (𝑂 evalF 𝑆)
16 yoneda.z . . . . 5 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
17 yoneda.c . . . . 5 (𝜑𝐶 ∈ Cat)
18 yoneda.w . . . . 5 (𝜑𝑉𝑊)
19 yoneda.u . . . . 5 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
20 yoneda.v . . . . 5 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
2110, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 17, 18, 19, 20yonedalem1 18317 . . . 4 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
2221simpld 494 . . 3 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
2321simprd 495 . . 3 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
24 yonedainv.i . . 3 𝐼 = (Inv‘𝑅)
25 eqid 2737 . . 3 (Inv‘𝑇) = (Inv‘𝑇)
26 yoneda.m . . . 4 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
2710, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 17, 18, 19, 20, 26yonedalem3 18325 . . 3 (𝜑𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸))
2817adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → 𝐶 ∈ Cat)
2918adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → 𝑉𝑊)
3019adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ran (Homf𝐶) ⊆ 𝑈)
3120adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
32 simprl 771 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ∈ (𝑂 Func 𝑆))
33 simprr 773 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → 𝑤𝐵)
3410, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 28, 29, 30, 31, 32, 33, 26yonedalem3a 18319 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤) = (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤))) ∧ (𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤)))
3534simprd 495 . . . . . . . . 9 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤))
3628adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → 𝐶 ∈ Cat)
3729adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → 𝑉𝑊)
3830adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → ran (Homf𝐶) ⊆ 𝑈)
3931adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
40 simplrl 777 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → ∈ (𝑂 Func 𝑆))
41 simplrr 778 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → 𝑤𝐵)
42 yonedainv.n . . . . . . . . . . . 12 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
43 simpr 484 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → 𝑏 ∈ ((1st)‘𝑤))
4410, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 36, 37, 38, 39, 40, 41, 42, 43yonedalem4c 18322 . . . . . . . . . . 11 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → ((𝑁𝑤)‘𝑏) ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
4544fmpttd 7135 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑏 ∈ ((1st)‘𝑤) ↦ ((𝑁𝑤)‘𝑏)):((1st)‘𝑤)⟶(((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
466fvexi 6920 . . . . . . . . . . . . . . 15 𝐵 ∈ V
4746mptex 7243 . . . . . . . . . . . . . 14 (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢))) ∈ V
48 eqid 2737 . . . . . . . . . . . . . 14 (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))) = (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢))))
4947, 48fnmpti 6711 . . . . . . . . . . . . 13 (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))) Fn ((1st)‘𝑤)
50 simpl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑓 = 𝑥 = 𝑤) → 𝑓 = )
5150fveq2d 6910 . . . . . . . . . . . . . . . . . 18 ((𝑓 = 𝑥 = 𝑤) → (1st𝑓) = (1st))
52 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝑓 = 𝑥 = 𝑤) → 𝑥 = 𝑤)
5351, 52fveq12d 6913 . . . . . . . . . . . . . . . . 17 ((𝑓 = 𝑥 = 𝑤) → ((1st𝑓)‘𝑥) = ((1st)‘𝑤))
54 simplr 769 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → 𝑥 = 𝑤)
5554oveq2d 7447 . . . . . . . . . . . . . . . . . . 19 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → (𝑦(Hom ‘𝐶)𝑥) = (𝑦(Hom ‘𝐶)𝑤))
56 simpll 767 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → 𝑓 = )
5756fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → (2nd𝑓) = (2nd))
58 eqidd 2738 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → 𝑦 = 𝑦)
5957, 54, 58oveq123d 7452 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → (𝑥(2nd𝑓)𝑦) = (𝑤(2nd)𝑦))
6059fveq1d 6908 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → ((𝑥(2nd𝑓)𝑦)‘𝑔) = ((𝑤(2nd)𝑦)‘𝑔))
6160fveq1d 6908 . . . . . . . . . . . . . . . . . . 19 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢) = (((𝑤(2nd)𝑦)‘𝑔)‘𝑢))
6255, 61mpteq12dv 5233 . . . . . . . . . . . . . . . . . 18 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)) = (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))
6362mpteq2dva 5242 . . . . . . . . . . . . . . . . 17 ((𝑓 = 𝑥 = 𝑤) → (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢))) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢))))
6453, 63mpteq12dv 5233 . . . . . . . . . . . . . . . 16 ((𝑓 = 𝑥 = 𝑤) → (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))) = (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))))
65 fvex 6919 . . . . . . . . . . . . . . . . 17 ((1st)‘𝑤) ∈ V
6665mptex 7243 . . . . . . . . . . . . . . . 16 (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))) ∈ V
6764, 42, 66ovmpoa 7588 . . . . . . . . . . . . . . 15 (( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵) → (𝑁𝑤) = (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))))
6867adantl 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑁𝑤) = (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))))
6968fneq1d 6661 . . . . . . . . . . . . 13 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑁𝑤) Fn ((1st)‘𝑤) ↔ (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))) Fn ((1st)‘𝑤)))
7049, 69mpbiri 258 . . . . . . . . . . . 12 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑁𝑤) Fn ((1st)‘𝑤))
71 dffn5 6967 . . . . . . . . . . . 12 ((𝑁𝑤) Fn ((1st)‘𝑤) ↔ (𝑁𝑤) = (𝑏 ∈ ((1st)‘𝑤) ↦ ((𝑁𝑤)‘𝑏)))
7270, 71sylib 218 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑁𝑤) = (𝑏 ∈ ((1st)‘𝑤) ↦ ((𝑁𝑤)‘𝑏)))
735oppccat 17765 . . . . . . . . . . . . . 14 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
7417, 73syl 17 . . . . . . . . . . . . 13 (𝜑𝑂 ∈ Cat)
7574adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → 𝑂 ∈ Cat)
7620unssbd 4194 . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
7718, 76ssexd 5324 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ V)
7812setccat 18130 . . . . . . . . . . . . . 14 (𝑈 ∈ V → 𝑆 ∈ Cat)
7977, 78syl 17 . . . . . . . . . . . . 13 (𝜑𝑆 ∈ Cat)
8079adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → 𝑆 ∈ Cat)
8115, 75, 80, 7, 32, 33evlf1 18265 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((1st𝐸)𝑤) = ((1st)‘𝑤))
8210, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 28, 29, 30, 31, 32, 33yonedalem21 18318 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((1st𝑍)𝑤) = (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
8372, 81, 82feq123d 6725 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑁𝑤):((1st𝐸)𝑤)⟶((1st𝑍)𝑤) ↔ (𝑏 ∈ ((1st)‘𝑤) ↦ ((𝑁𝑤)‘𝑏)):((1st)‘𝑤)⟶(((1st𝑌)‘𝑤)(𝑂 Nat 𝑆))))
8445, 83mpbird 257 . . . . . . . . 9 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑁𝑤):((1st𝐸)𝑤)⟶((1st𝑍)𝑤))
85 fcompt 7153 . . . . . . . . . . 11 (((𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤) ∧ (𝑁𝑤):((1st𝐸)𝑤)⟶((1st𝑍)𝑤)) → ((𝑀𝑤) ∘ (𝑁𝑤)) = (𝑘 ∈ ((1st𝐸)𝑤) ↦ ((𝑀𝑤)‘((𝑁𝑤)‘𝑘))))
8635, 84, 85syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤) ∘ (𝑁𝑤)) = (𝑘 ∈ ((1st𝐸)𝑤) ↦ ((𝑀𝑤)‘((𝑁𝑤)‘𝑘))))
8781eleq2d 2827 . . . . . . . . . . . . . 14 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑘 ∈ ((1st𝐸)𝑤) ↔ 𝑘 ∈ ((1st)‘𝑤)))
8887biimpa 476 . . . . . . . . . . . . 13 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st𝐸)𝑤)) → 𝑘 ∈ ((1st)‘𝑤))
8928adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → 𝐶 ∈ Cat)
9029adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → 𝑉𝑊)
9130adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ran (Homf𝐶) ⊆ 𝑈)
9231adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
93 simplrl 777 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ∈ (𝑂 Func 𝑆))
94 simplrr 778 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → 𝑤𝐵)
9510, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 89, 90, 91, 92, 93, 94, 26yonedalem3a 18319 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑀𝑤) = (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤))) ∧ (𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤)))
9695simpld 494 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (𝑀𝑤) = (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤))))
9796fveq1d 6908 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑀𝑤)‘((𝑁𝑤)‘𝑘)) = ((𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))‘((𝑁𝑤)‘𝑘)))
9872, 44fmpt3d 7136 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑁𝑤):((1st)‘𝑤)⟶(((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
9998ffvelcdmda 7104 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑁𝑤)‘𝑘) ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
100 fveq1 6905 . . . . . . . . . . . . . . . . 17 (𝑎 = ((𝑁𝑤)‘𝑘) → (𝑎𝑤) = (((𝑁𝑤)‘𝑘)‘𝑤))
101100fveq1d 6908 . . . . . . . . . . . . . . . 16 (𝑎 = ((𝑁𝑤)‘𝑘) → ((𝑎𝑤)‘( 1𝑤)) = ((((𝑁𝑤)‘𝑘)‘𝑤)‘( 1𝑤)))
102 eqid 2737 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤))) = (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))
103 fvex 6919 . . . . . . . . . . . . . . . 16 ((((𝑁𝑤)‘𝑘)‘𝑤)‘( 1𝑤)) ∈ V
104101, 102, 103fvmpt 7016 . . . . . . . . . . . . . . 15 (((𝑁𝑤)‘𝑘) ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) → ((𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))‘((𝑁𝑤)‘𝑘)) = ((((𝑁𝑤)‘𝑘)‘𝑤)‘( 1𝑤)))
10599, 104syl 17 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))‘((𝑁𝑤)‘𝑘)) = ((((𝑁𝑤)‘𝑘)‘𝑤)‘( 1𝑤)))
106 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → 𝑘 ∈ ((1st)‘𝑤))
107 eqid 2737 . . . . . . . . . . . . . . . . 17 (Hom ‘𝐶) = (Hom ‘𝐶)
1086, 107, 11, 89, 94catidcl 17725 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ( 1𝑤) ∈ (𝑤(Hom ‘𝐶)𝑤))
10910, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 89, 90, 91, 92, 93, 94, 42, 106, 94, 108yonedalem4b 18321 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((((𝑁𝑤)‘𝑘)‘𝑤)‘( 1𝑤)) = (((𝑤(2nd)𝑤)‘( 1𝑤))‘𝑘))
110 eqid 2737 . . . . . . . . . . . . . . . . . 18 (Id‘𝑂) = (Id‘𝑂)
111 eqid 2737 . . . . . . . . . . . . . . . . . 18 (Id‘𝑆) = (Id‘𝑆)
112 relfunc 17907 . . . . . . . . . . . . . . . . . . 19 Rel (𝑂 Func 𝑆)
113 1st2ndbr 8067 . . . . . . . . . . . . . . . . . . 19 ((Rel (𝑂 Func 𝑆) ∧ ∈ (𝑂 Func 𝑆)) → (1st)(𝑂 Func 𝑆)(2nd))
114112, 93, 113sylancr 587 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (1st)(𝑂 Func 𝑆)(2nd))
1157, 110, 111, 114, 94funcid 17915 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑤(2nd)𝑤)‘((Id‘𝑂)‘𝑤)) = ((Id‘𝑆)‘((1st)‘𝑤)))
1165, 11oppcid 17764 . . . . . . . . . . . . . . . . . . . 20 (𝐶 ∈ Cat → (Id‘𝑂) = 1 )
11789, 116syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (Id‘𝑂) = 1 )
118117fveq1d 6908 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((Id‘𝑂)‘𝑤) = ( 1𝑤))
119118fveq2d 6910 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑤(2nd)𝑤)‘((Id‘𝑂)‘𝑤)) = ((𝑤(2nd)𝑤)‘( 1𝑤)))
12077ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → 𝑈 ∈ V)
121 eqid 2737 . . . . . . . . . . . . . . . . . . . . 21 (Base‘𝑆) = (Base‘𝑆)
1227, 121, 114funcf1 17911 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (1st):𝐵⟶(Base‘𝑆))
12312, 120setcbas 18123 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → 𝑈 = (Base‘𝑆))
124123feq3d 6723 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((1st):𝐵𝑈 ↔ (1st):𝐵⟶(Base‘𝑆)))
125122, 124mpbird 257 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (1st):𝐵𝑈)
126125, 94ffvelcdmd 7105 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((1st)‘𝑤) ∈ 𝑈)
12712, 111, 120, 126setcid 18131 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((Id‘𝑆)‘((1st)‘𝑤)) = ( I ↾ ((1st)‘𝑤)))
128115, 119, 1273eqtr3d 2785 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑤(2nd)𝑤)‘( 1𝑤)) = ( I ↾ ((1st)‘𝑤)))
129128fveq1d 6908 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (((𝑤(2nd)𝑤)‘( 1𝑤))‘𝑘) = (( I ↾ ((1st)‘𝑤))‘𝑘))
130 fvresi 7193 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ((1st)‘𝑤) → (( I ↾ ((1st)‘𝑤))‘𝑘) = 𝑘)
131130adantl 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (( I ↾ ((1st)‘𝑤))‘𝑘) = 𝑘)
132109, 129, 1313eqtrd 2781 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((((𝑁𝑤)‘𝑘)‘𝑤)‘( 1𝑤)) = 𝑘)
13397, 105, 1323eqtrd 2781 . . . . . . . . . . . . 13 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑀𝑤)‘((𝑁𝑤)‘𝑘)) = 𝑘)
13488, 133syldan 591 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st𝐸)𝑤)) → ((𝑀𝑤)‘((𝑁𝑤)‘𝑘)) = 𝑘)
135134mpteq2dva 5242 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑘 ∈ ((1st𝐸)𝑤) ↦ ((𝑀𝑤)‘((𝑁𝑤)‘𝑘))) = (𝑘 ∈ ((1st𝐸)𝑤) ↦ 𝑘))
136 mptresid 6069 . . . . . . . . . . 11 ( I ↾ ((1st𝐸)𝑤)) = (𝑘 ∈ ((1st𝐸)𝑤) ↦ 𝑘)
137135, 136eqtr4di 2795 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑘 ∈ ((1st𝐸)𝑤) ↦ ((𝑀𝑤)‘((𝑁𝑤)‘𝑘))) = ( I ↾ ((1st𝐸)𝑤)))
13886, 137eqtrd 2777 . . . . . . . . 9 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤) ∘ (𝑁𝑤)) = ( I ↾ ((1st𝐸)𝑤)))
139 fcompt 7153 . . . . . . . . . . 11 (((𝑁𝑤):((1st𝐸)𝑤)⟶((1st𝑍)𝑤) ∧ (𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤)) → ((𝑁𝑤) ∘ (𝑀𝑤)) = (𝑏 ∈ ((1st𝑍)𝑤) ↦ ((𝑁𝑤)‘((𝑀𝑤)‘𝑏))))
14084, 35, 139syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑁𝑤) ∘ (𝑀𝑤)) = (𝑏 ∈ ((1st𝑍)𝑤) ↦ ((𝑁𝑤)‘((𝑀𝑤)‘𝑏))))
141 eqid 2737 . . . . . . . . . . . . . 14 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
14228adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝐶 ∈ Cat)
14329adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑉𝑊)
14430adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ran (Homf𝐶) ⊆ 𝑈)
14531adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
146 simplrl 777 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ∈ (𝑂 Func 𝑆))
147 simplrr 778 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑤𝐵)
14881feq3d 6723 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤) ↔ (𝑀𝑤):((1st𝑍)𝑤)⟶((1st)‘𝑤)))
14935, 148mpbid 232 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑀𝑤):((1st𝑍)𝑤)⟶((1st)‘𝑤))
150149ffvelcdmda 7104 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((𝑀𝑤)‘𝑏) ∈ ((1st)‘𝑤))
15110, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 142, 143, 144, 145, 146, 147, 42, 150yonedalem4c 18322 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((𝑁𝑤)‘((𝑀𝑤)‘𝑏)) ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
152141, 151nat1st2nd 17999 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((𝑁𝑤)‘((𝑀𝑤)‘𝑏)) ∈ (⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩(𝑂 Nat 𝑆)⟨(1st), (2nd)⟩))
153141, 152, 7natfn 18002 . . . . . . . . . . . . 13 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((𝑁𝑤)‘((𝑀𝑤)‘𝑏)) Fn 𝐵)
15482eleq2d 2827 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑏 ∈ ((1st𝑍)𝑤) ↔ 𝑏 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆))))
155154biimpa 476 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑏 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
156141, 155nat1st2nd 17999 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑏 ∈ (⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩(𝑂 Nat 𝑆)⟨(1st), (2nd)⟩))
157141, 156, 7natfn 18002 . . . . . . . . . . . . 13 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑏 Fn 𝐵)
158142adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → 𝐶 ∈ Cat)
159147adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → 𝑤𝐵)
160 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → 𝑧𝐵)
16110, 6, 158, 159, 107, 160yon11 18309 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) = (𝑧(Hom ‘𝐶)𝑤))
162161eleq2d 2827 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ↔ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))
163162biimpa 476 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤))
164158adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝐶 ∈ Cat)
165143ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑉𝑊)
166144ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ran (Homf𝐶) ⊆ 𝑈)
167145ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
168146ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ∈ (𝑂 Func 𝑆))
169159adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑤𝐵)
170150ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑀𝑤)‘𝑏) ∈ ((1st)‘𝑤))
171 simplr 769 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑧𝐵)
172 simpr 484 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤))
17310, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 164, 165, 166, 167, 168, 169, 42, 170, 171, 172yonedalem4b 18321 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧)‘𝑘) = (((𝑤(2nd)𝑧)‘𝑘)‘((𝑀𝑤)‘𝑏)))
17410, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 164, 165, 166, 167, 168, 169, 26yonedalem3a 18319 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑀𝑤) = (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤))) ∧ (𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤)))
175174simpld 494 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑀𝑤) = (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤))))
176175fveq1d 6908 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑀𝑤)‘𝑏) = ((𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))‘𝑏))
177155ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑏 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
178 fveq1 6905 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑏 → (𝑎𝑤) = (𝑏𝑤))
179178fveq1d 6908 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 𝑏 → ((𝑎𝑤)‘( 1𝑤)) = ((𝑏𝑤)‘( 1𝑤)))
180 fvex 6919 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏𝑤)‘( 1𝑤)) ∈ V
181179, 102, 180fvmpt 7016 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) → ((𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))‘𝑏) = ((𝑏𝑤)‘( 1𝑤)))
182177, 181syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))‘𝑏) = ((𝑏𝑤)‘( 1𝑤)))
183176, 182eqtrd 2777 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑀𝑤)‘𝑏) = ((𝑏𝑤)‘( 1𝑤)))
184183fveq2d 6910 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd)𝑧)‘𝑘)‘((𝑀𝑤)‘𝑏)) = (((𝑤(2nd)𝑧)‘𝑘)‘((𝑏𝑤)‘( 1𝑤))))
185156ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑏 ∈ (⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩(𝑂 Nat 𝑆)⟨(1st), (2nd)⟩))
186 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 22 (Hom ‘𝑂) = (Hom ‘𝑂)
187 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 22 (comp‘𝑆) = (comp‘𝑆)
188107, 5oppchom 17758 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑤)
189172, 188eleqtrrdi 2852 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑘 ∈ (𝑤(Hom ‘𝑂)𝑧))
190141, 185, 7, 186, 187, 169, 171, 189nati 18003 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏𝑧)(⟨((1st ‘((1st𝑌)‘𝑤))‘𝑤), ((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟩(comp‘𝑆)((1st)‘𝑧))((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)) = (((𝑤(2nd)𝑧)‘𝑘)(⟨((1st ‘((1st𝑌)‘𝑤))‘𝑤), ((1st)‘𝑤)⟩(comp‘𝑆)((1st)‘𝑧))(𝑏𝑤)))
19177ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑈 ∈ V)
192191adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → 𝑈 ∈ V)
193192adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑈 ∈ V)
194 relfunc 17907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Rel (𝐶 Func 𝑄)
19510, 17, 5, 12, 3, 77, 19yoncl 18307 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
196 1st2ndbr 8067 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
197194, 195, 196sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
1986, 4, 197funcf1 17911 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
199198ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
200199, 147ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
201 1st2ndbr 8067 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑤)))
202112, 200, 201sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st ‘((1st𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑤)))
2037, 121, 202funcf1 17911 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st ‘((1st𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
20412, 191setcbas 18123 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑈 = (Base‘𝑆))
205204feq3d 6723 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((1st ‘((1st𝑌)‘𝑤)):𝐵𝑈 ↔ (1st ‘((1st𝑌)‘𝑤)):𝐵⟶(Base‘𝑆)))
206203, 205mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st ‘((1st𝑌)‘𝑤)):𝐵𝑈)
207206, 147ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑤) ∈ 𝑈)
208207ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑤) ∈ 𝑈)
209206ffvelcdmda 7104 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ∈ 𝑈)
210209adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ∈ 𝑈)
211112, 146, 113sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st)(𝑂 Func 𝑆)(2nd))
2127, 121, 211funcf1 17911 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st):𝐵⟶(Base‘𝑆))
213204feq3d 6723 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((1st):𝐵𝑈 ↔ (1st):𝐵⟶(Base‘𝑆)))
214212, 213mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st):𝐵𝑈)
215214ffvelcdmda 7104 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → ((1st)‘𝑧) ∈ 𝑈)
216215adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st)‘𝑧) ∈ 𝑈)
217 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Hom ‘𝑆) = (Hom ‘𝑆)
218202ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑤)))
2197, 186, 217, 218, 169, 171funcf2 17913 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧):(𝑤(Hom ‘𝑂)𝑧)⟶(((1st ‘((1st𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑤))‘𝑧)))
220219, 189ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑤))‘𝑧)))
22112, 193, 217, 208, 210elsetchom 18126 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑤))‘𝑧)) ↔ ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘):((1st ‘((1st𝑌)‘𝑤))‘𝑤)⟶((1st ‘((1st𝑌)‘𝑤))‘𝑧)))
222220, 221mpbid 232 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘):((1st ‘((1st𝑌)‘𝑤))‘𝑤)⟶((1st ‘((1st𝑌)‘𝑤))‘𝑧))
223156adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → 𝑏 ∈ (⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩(𝑂 Nat 𝑆)⟨(1st), (2nd)⟩))
224141, 223, 7, 217, 160natcl 18001 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (𝑏𝑧) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑧)(Hom ‘𝑆)((1st)‘𝑧)))
22512, 192, 217, 209, 215elsetchom 18126 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → ((𝑏𝑧) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑧)(Hom ‘𝑆)((1st)‘𝑧)) ↔ (𝑏𝑧):((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟶((1st)‘𝑧)))
226224, 225mpbid 232 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (𝑏𝑧):((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟶((1st)‘𝑧))
227226adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑏𝑧):((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟶((1st)‘𝑧))
22812, 193, 187, 208, 210, 216, 222, 227setcco 18128 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏𝑧)(⟨((1st ‘((1st𝑌)‘𝑤))‘𝑤), ((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟩(comp‘𝑆)((1st)‘𝑧))((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)) = ((𝑏𝑧) ∘ ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)))
229214, 147ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((1st)‘𝑤) ∈ 𝑈)
230229ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st)‘𝑤) ∈ 𝑈)
231141, 156, 7, 217, 147natcl 18001 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (𝑏𝑤) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st)‘𝑤)))
23212, 191, 217, 207, 229elsetchom 18126 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((𝑏𝑤) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st)‘𝑤)) ↔ (𝑏𝑤):((1st ‘((1st𝑌)‘𝑤))‘𝑤)⟶((1st)‘𝑤)))
233231, 232mpbid 232 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (𝑏𝑤):((1st ‘((1st𝑌)‘𝑤))‘𝑤)⟶((1st)‘𝑤))
234233ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑏𝑤):((1st ‘((1st𝑌)‘𝑤))‘𝑤)⟶((1st)‘𝑤))
235112, 168, 113sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st)(𝑂 Func 𝑆)(2nd))
2367, 186, 217, 235, 169, 171funcf2 17913 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑤(2nd)𝑧):(𝑤(Hom ‘𝑂)𝑧)⟶(((1st)‘𝑤)(Hom ‘𝑆)((1st)‘𝑧)))
237236, 189ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑤(2nd)𝑧)‘𝑘) ∈ (((1st)‘𝑤)(Hom ‘𝑆)((1st)‘𝑧)))
23812, 193, 217, 230, 216elsetchom 18126 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd)𝑧)‘𝑘) ∈ (((1st)‘𝑤)(Hom ‘𝑆)((1st)‘𝑧)) ↔ ((𝑤(2nd)𝑧)‘𝑘):((1st)‘𝑤)⟶((1st)‘𝑧)))
239237, 238mpbid 232 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑤(2nd)𝑧)‘𝑘):((1st)‘𝑤)⟶((1st)‘𝑧))
24012, 193, 187, 208, 230, 216, 234, 239setcco 18128 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd)𝑧)‘𝑘)(⟨((1st ‘((1st𝑌)‘𝑤))‘𝑤), ((1st)‘𝑤)⟩(comp‘𝑆)((1st)‘𝑧))(𝑏𝑤)) = (((𝑤(2nd)𝑧)‘𝑘) ∘ (𝑏𝑤)))
241190, 228, 2403eqtr3d 2785 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏𝑧) ∘ ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)) = (((𝑤(2nd)𝑧)‘𝑘) ∘ (𝑏𝑤)))
242241fveq1d 6908 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑏𝑧) ∘ ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘))‘( 1𝑤)) = ((((𝑤(2nd)𝑧)‘𝑘) ∘ (𝑏𝑤))‘( 1𝑤)))
2436, 107, 11, 142, 147catidcl 17725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ( 1𝑤) ∈ (𝑤(Hom ‘𝐶)𝑤))
24410, 6, 142, 147, 107, 147yon11 18309 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑤) = (𝑤(Hom ‘𝐶)𝑤))
245243, 244eleqtrrd 2844 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ( 1𝑤) ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑤))
246245ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ( 1𝑤) ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑤))
247222, 246fvco3d 7009 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑏𝑧) ∘ ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘))‘( 1𝑤)) = ((𝑏𝑧)‘(((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)‘( 1𝑤))))
248233, 245fvco3d 7009 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((((𝑤(2nd)𝑧)‘𝑘) ∘ (𝑏𝑤))‘( 1𝑤)) = (((𝑤(2nd)𝑧)‘𝑘)‘((𝑏𝑤)‘( 1𝑤))))
249248ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((𝑤(2nd)𝑧)‘𝑘) ∘ (𝑏𝑤))‘( 1𝑤)) = (((𝑤(2nd)𝑧)‘𝑘)‘((𝑏𝑤)‘( 1𝑤))))
250242, 247, 2493eqtr3d 2785 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏𝑧)‘(((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)‘( 1𝑤))) = (((𝑤(2nd)𝑧)‘𝑘)‘((𝑏𝑤)‘( 1𝑤))))
251 eqid 2737 . . . . . . . . . . . . . . . . . . . . 21 (comp‘𝐶) = (comp‘𝐶)
252243ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ( 1𝑤) ∈ (𝑤(Hom ‘𝐶)𝑤))
25310, 6, 164, 169, 107, 169, 251, 171, 172, 252yon12 18310 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)‘( 1𝑤)) = (( 1𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐶)𝑤)𝑘))
2546, 107, 11, 164, 171, 251, 169, 172catlid 17726 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (( 1𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐶)𝑤)𝑘) = 𝑘)
255253, 254eqtrd 2777 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)‘( 1𝑤)) = 𝑘)
256255fveq2d 6910 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏𝑧)‘(((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)‘( 1𝑤))) = ((𝑏𝑧)‘𝑘))
257250, 256eqtr3d 2779 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd)𝑧)‘𝑘)‘((𝑏𝑤)‘( 1𝑤))) = ((𝑏𝑧)‘𝑘))
258173, 184, 2573eqtrd 2781 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧)‘𝑘) = ((𝑏𝑧)‘𝑘))
259163, 258syldan 591 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧)) → ((((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧)‘𝑘) = ((𝑏𝑧)‘𝑘))
260259mpteq2dva 5242 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ↦ ((((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧)‘𝑘)) = (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ↦ ((𝑏𝑧)‘𝑘)))
261152adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → ((𝑁𝑤)‘((𝑀𝑤)‘𝑏)) ∈ (⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩(𝑂 Nat 𝑆)⟨(1st), (2nd)⟩))
262141, 261, 7, 217, 160natcl 18001 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑧)(Hom ‘𝑆)((1st)‘𝑧)))
26312, 192, 217, 209, 215elsetchom 18126 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → ((((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑧)(Hom ‘𝑆)((1st)‘𝑧)) ↔ (((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧):((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟶((1st)‘𝑧)))
264262, 263mpbid 232 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧):((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟶((1st)‘𝑧))
265264feqmptd 6977 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧) = (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ↦ ((((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧)‘𝑘)))
266226feqmptd 6977 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (𝑏𝑧) = (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ↦ ((𝑏𝑧)‘𝑘)))
267260, 265, 2663eqtr4d 2787 . . . . . . . . . . . . 13 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧) = (𝑏𝑧))
268153, 157, 267eqfnfvd 7054 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((𝑁𝑤)‘((𝑀𝑤)‘𝑏)) = 𝑏)
269268mpteq2dva 5242 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑏 ∈ ((1st𝑍)𝑤) ↦ ((𝑁𝑤)‘((𝑀𝑤)‘𝑏))) = (𝑏 ∈ ((1st𝑍)𝑤) ↦ 𝑏))
270 mptresid 6069 . . . . . . . . . . 11 ( I ↾ ((1st𝑍)𝑤)) = (𝑏 ∈ ((1st𝑍)𝑤) ↦ 𝑏)
271269, 270eqtr4di 2795 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑏 ∈ ((1st𝑍)𝑤) ↦ ((𝑁𝑤)‘((𝑀𝑤)‘𝑏))) = ( I ↾ ((1st𝑍)𝑤)))
272140, 271eqtrd 2777 . . . . . . . . 9 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑁𝑤) ∘ (𝑀𝑤)) = ( I ↾ ((1st𝑍)𝑤)))
273 fcof1o 7316 . . . . . . . . 9 ((((𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤) ∧ (𝑁𝑤):((1st𝐸)𝑤)⟶((1st𝑍)𝑤)) ∧ (((𝑀𝑤) ∘ (𝑁𝑤)) = ( I ↾ ((1st𝐸)𝑤)) ∧ ((𝑁𝑤) ∘ (𝑀𝑤)) = ( I ↾ ((1st𝑍)𝑤)))) → ((𝑀𝑤):((1st𝑍)𝑤)–1-1-onto→((1st𝐸)𝑤) ∧ (𝑀𝑤) = (𝑁𝑤)))
27435, 84, 138, 272, 273syl22anc 839 . . . . . . . 8 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤):((1st𝑍)𝑤)–1-1-onto→((1st𝐸)𝑤) ∧ (𝑀𝑤) = (𝑁𝑤)))
275 eqcom 2744 . . . . . . . . 9 ((𝑀𝑤) = (𝑁𝑤) ↔ (𝑁𝑤) = (𝑀𝑤))
276275anbi2i 623 . . . . . . . 8 (((𝑀𝑤):((1st𝑍)𝑤)–1-1-onto→((1st𝐸)𝑤) ∧ (𝑀𝑤) = (𝑁𝑤)) ↔ ((𝑀𝑤):((1st𝑍)𝑤)–1-1-onto→((1st𝐸)𝑤) ∧ (𝑁𝑤) = (𝑀𝑤)))
277274, 276sylib 218 . . . . . . 7 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤):((1st𝑍)𝑤)–1-1-onto→((1st𝐸)𝑤) ∧ (𝑁𝑤) = (𝑀𝑤)))
278 eqid 2737 . . . . . . . . . . 11 (Base‘𝑇) = (Base‘𝑇)
279 relfunc 17907 . . . . . . . . . . . 12 Rel ((𝑄 ×c 𝑂) Func 𝑇)
280 1st2ndbr 8067 . . . . . . . . . . . 12 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
281279, 22, 280sylancr 587 . . . . . . . . . . 11 (𝜑 → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
2828, 278, 281funcf1 17911 . . . . . . . . . 10 (𝜑 → (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
28313, 18setcbas 18123 . . . . . . . . . . 11 (𝜑𝑉 = (Base‘𝑇))
284283feq3d 6723 . . . . . . . . . 10 (𝜑 → ((1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶𝑉 ↔ (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇)))
285282, 284mpbird 257 . . . . . . . . 9 (𝜑 → (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶𝑉)
286285fovcdmda 7604 . . . . . . . 8 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((1st𝑍)𝑤) ∈ 𝑉)
287 1st2ndbr 8067 . . . . . . . . . . . 12 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
288279, 23, 287sylancr 587 . . . . . . . . . . 11 (𝜑 → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
2898, 278, 288funcf1 17911 . . . . . . . . . 10 (𝜑 → (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
290283feq3d 6723 . . . . . . . . . 10 (𝜑 → ((1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶𝑉 ↔ (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇)))
291289, 290mpbird 257 . . . . . . . . 9 (𝜑 → (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶𝑉)
292291fovcdmda 7604 . . . . . . . 8 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((1st𝐸)𝑤) ∈ 𝑉)
29313, 29, 286, 292, 25setcinv 18135 . . . . . . 7 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤)(((1st𝑍)𝑤)(Inv‘𝑇)((1st𝐸)𝑤))(𝑁𝑤) ↔ ((𝑀𝑤):((1st𝑍)𝑤)–1-1-onto→((1st𝐸)𝑤) ∧ (𝑁𝑤) = (𝑀𝑤))))
294277, 293mpbird 257 . . . . . 6 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑀𝑤)(((1st𝑍)𝑤)(Inv‘𝑇)((1st𝐸)𝑤))(𝑁𝑤))
295294ralrimivva 3202 . . . . 5 (𝜑 → ∀ ∈ (𝑂 Func 𝑆)∀𝑤𝐵 (𝑀𝑤)(((1st𝑍)𝑤)(Inv‘𝑇)((1st𝐸)𝑤))(𝑁𝑤))
296 fveq2 6906 . . . . . . . 8 (𝑧 = ⟨, 𝑤⟩ → (𝑀𝑧) = (𝑀‘⟨, 𝑤⟩))
297 df-ov 7434 . . . . . . . 8 (𝑀𝑤) = (𝑀‘⟨, 𝑤⟩)
298296, 297eqtr4di 2795 . . . . . . 7 (𝑧 = ⟨, 𝑤⟩ → (𝑀𝑧) = (𝑀𝑤))
299 fveq2 6906 . . . . . . . . 9 (𝑧 = ⟨, 𝑤⟩ → ((1st𝑍)‘𝑧) = ((1st𝑍)‘⟨, 𝑤⟩))
300 df-ov 7434 . . . . . . . . 9 ((1st𝑍)𝑤) = ((1st𝑍)‘⟨, 𝑤⟩)
301299, 300eqtr4di 2795 . . . . . . . 8 (𝑧 = ⟨, 𝑤⟩ → ((1st𝑍)‘𝑧) = ((1st𝑍)𝑤))
302 fveq2 6906 . . . . . . . . 9 (𝑧 = ⟨, 𝑤⟩ → ((1st𝐸)‘𝑧) = ((1st𝐸)‘⟨, 𝑤⟩))
303 df-ov 7434 . . . . . . . . 9 ((1st𝐸)𝑤) = ((1st𝐸)‘⟨, 𝑤⟩)
304302, 303eqtr4di 2795 . . . . . . . 8 (𝑧 = ⟨, 𝑤⟩ → ((1st𝐸)‘𝑧) = ((1st𝐸)𝑤))
305301, 304oveq12d 7449 . . . . . . 7 (𝑧 = ⟨, 𝑤⟩ → (((1st𝑍)‘𝑧)(Inv‘𝑇)((1st𝐸)‘𝑧)) = (((1st𝑍)𝑤)(Inv‘𝑇)((1st𝐸)𝑤)))
306 fveq2 6906 . . . . . . . 8 (𝑧 = ⟨, 𝑤⟩ → (𝑁𝑧) = (𝑁‘⟨, 𝑤⟩))
307 df-ov 7434 . . . . . . . 8 (𝑁𝑤) = (𝑁‘⟨, 𝑤⟩)
308306, 307eqtr4di 2795 . . . . . . 7 (𝑧 = ⟨, 𝑤⟩ → (𝑁𝑧) = (𝑁𝑤))
309298, 305, 308breq123d 5157 . . . . . 6 (𝑧 = ⟨, 𝑤⟩ → ((𝑀𝑧)(((1st𝑍)‘𝑧)(Inv‘𝑇)((1st𝐸)‘𝑧))(𝑁𝑧) ↔ (𝑀𝑤)(((1st𝑍)𝑤)(Inv‘𝑇)((1st𝐸)𝑤))(𝑁𝑤)))
310309ralxp 5852 . . . . 5 (∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀𝑧)(((1st𝑍)‘𝑧)(Inv‘𝑇)((1st𝐸)‘𝑧))(𝑁𝑧) ↔ ∀ ∈ (𝑂 Func 𝑆)∀𝑤𝐵 (𝑀𝑤)(((1st𝑍)𝑤)(Inv‘𝑇)((1st𝐸)𝑤))(𝑁𝑤))
311295, 310sylibr 234 . . . 4 (𝜑 → ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀𝑧)(((1st𝑍)‘𝑧)(Inv‘𝑇)((1st𝐸)‘𝑧))(𝑁𝑧))
312311r19.21bi 3251 . . 3 ((𝜑𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)) → (𝑀𝑧)(((1st𝑍)‘𝑧)(Inv‘𝑇)((1st𝐸)‘𝑧))(𝑁𝑧))
3131, 8, 9, 22, 23, 24, 25, 27, 312invfuc 18022 . 2 (𝜑𝑀(𝑍𝐼𝐸)(𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ↦ (𝑁𝑧)))
314 fvex 6919 . . . . 5 ((1st𝑓)‘𝑥) ∈ V
315314mptex 7243 . . . 4 (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))) ∈ V
31642, 315fnmpoi 8095 . . 3 𝑁 Fn ((𝑂 Func 𝑆) × 𝐵)
317 dffn5 6967 . . 3 (𝑁 Fn ((𝑂 Func 𝑆) × 𝐵) ↔ 𝑁 = (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ↦ (𝑁𝑧)))
318316, 317mpbi 230 . 2 𝑁 = (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ↦ (𝑁𝑧))
319313, 318breqtrrdi 5185 1 (𝜑𝑀(𝑍𝐼𝐸)𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  cun 3949  wss 3951  cop 4632   class class class wbr 5143  cmpt 5225   I cid 5577   × cxp 5683  ccnv 5684  ran crn 5686  cres 5687  ccom 5689  Rel wrel 5690   Fn wfn 6556  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  tpos ctpos 8250  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Idccid 17708  Homf chomf 17709  oppCatcoppc 17754  Invcinv 17789   Func cfunc 17899  func ccofu 17901   Nat cnat 17989   FuncCat cfuc 17990  SetCatcsetc 18120   ×c cxpc 18213   1stF c1stf 18214   2ndF c2ndf 18215   ⟨,⟩F cprf 18216   evalF cevlf 18254  HomFchof 18293  Yoncyon 18294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-hom 17321  df-cco 17322  df-cat 17711  df-cid 17712  df-homf 17713  df-comf 17714  df-oppc 17755  df-sect 17791  df-inv 17792  df-ssc 17854  df-resc 17855  df-subc 17856  df-func 17903  df-cofu 17905  df-nat 17991  df-fuc 17992  df-setc 18121  df-xpc 18217  df-1stf 18218  df-2ndf 18219  df-prf 18220  df-evlf 18258  df-curf 18259  df-hof 18295  df-yon 18296
This theorem is referenced by:  yonffthlem  18327  yoneda  18328
  Copyright terms: Public domain W3C validator