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Theorem catcisolem 18155
Description: Lemma for catciso 18156. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
catciso.c 𝐶 = (CatCat‘𝑈)
catciso.b 𝐵 = (Base‘𝐶)
catciso.r 𝑅 = (Base‘𝑋)
catciso.s 𝑆 = (Base‘𝑌)
catciso.u (𝜑𝑈𝑉)
catciso.x (𝜑𝑋𝐵)
catciso.y (𝜑𝑌𝐵)
catcisolem.i 𝐼 = (Inv‘𝐶)
catcisolem.g 𝐻 = (𝑥𝑆, 𝑦𝑆((𝐹𝑥)𝐺(𝐹𝑦)))
catcisolem.1 (𝜑𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
catcisolem.2 (𝜑𝐹:𝑅1-1-onto𝑆)
Assertion
Ref Expression
catcisolem (𝜑 → ⟨𝐹, 𝐺⟩(𝑋𝐼𝑌)⟨𝐹, 𝐻⟩)
Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦   𝑥,𝐼,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem catcisolem
Dummy variables 𝑓 𝑔 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcisolem.2 . . . . . . 7 (𝜑𝐹:𝑅1-1-onto𝑆)
2 f1ococnv1 6877 . . . . . . 7 (𝐹:𝑅1-1-onto𝑆 → (𝐹𝐹) = ( I ↾ 𝑅))
31, 2syl 17 . . . . . 6 (𝜑 → (𝐹𝐹) = ( I ↾ 𝑅))
413ad2ant1 1134 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑅𝑣𝑅) → 𝐹:𝑅1-1-onto𝑆)
5 f1of 6848 . . . . . . . . . . . . . 14 (𝐹:𝑅1-1-onto𝑆𝐹:𝑅𝑆)
64, 5syl 17 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → 𝐹:𝑅𝑆)
7 simp2 1138 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → 𝑢𝑅)
86, 7ffvelcdmd 7105 . . . . . . . . . . . 12 ((𝜑𝑢𝑅𝑣𝑅) → (𝐹𝑢) ∈ 𝑆)
9 simp3 1139 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → 𝑣𝑅)
106, 9ffvelcdmd 7105 . . . . . . . . . . . 12 ((𝜑𝑢𝑅𝑣𝑅) → (𝐹𝑣) ∈ 𝑆)
11 simpl 482 . . . . . . . . . . . . . . . 16 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → 𝑥 = (𝐹𝑢))
1211fveq2d 6910 . . . . . . . . . . . . . . 15 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → (𝐹𝑥) = (𝐹‘(𝐹𝑢)))
13 simpr 484 . . . . . . . . . . . . . . . 16 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → 𝑦 = (𝐹𝑣))
1413fveq2d 6910 . . . . . . . . . . . . . . 15 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → (𝐹𝑦) = (𝐹‘(𝐹𝑣)))
1512, 14oveq12d 7449 . . . . . . . . . . . . . 14 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))))
1615cnveqd 5886 . . . . . . . . . . . . 13 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))))
17 catcisolem.g . . . . . . . . . . . . 13 𝐻 = (𝑥𝑆, 𝑦𝑆((𝐹𝑥)𝐺(𝐹𝑦)))
18 ovex 7464 . . . . . . . . . . . . . 14 ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))) ∈ V
1918cnvex 7947 . . . . . . . . . . . . 13 ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))) ∈ V
2016, 17, 19ovmpoa 7588 . . . . . . . . . . . 12 (((𝐹𝑢) ∈ 𝑆 ∧ (𝐹𝑣) ∈ 𝑆) → ((𝐹𝑢)𝐻(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))))
218, 10, 20syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑢𝑅𝑣𝑅) → ((𝐹𝑢)𝐻(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))))
22 f1ocnvfv1 7296 . . . . . . . . . . . . . 14 ((𝐹:𝑅1-1-onto𝑆𝑢𝑅) → (𝐹‘(𝐹𝑢)) = 𝑢)
234, 7, 22syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → (𝐹‘(𝐹𝑢)) = 𝑢)
24 f1ocnvfv1 7296 . . . . . . . . . . . . . 14 ((𝐹:𝑅1-1-onto𝑆𝑣𝑅) → (𝐹‘(𝐹𝑣)) = 𝑣)
254, 9, 24syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → (𝐹‘(𝐹𝑣)) = 𝑣)
2623, 25oveq12d 7449 . . . . . . . . . . . 12 ((𝜑𝑢𝑅𝑣𝑅) → ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))) = (𝑢𝐺𝑣))
2726cnveqd 5886 . . . . . . . . . . 11 ((𝜑𝑢𝑅𝑣𝑅) → ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))) = (𝑢𝐺𝑣))
2821, 27eqtrd 2777 . . . . . . . . . 10 ((𝜑𝑢𝑅𝑣𝑅) → ((𝐹𝑢)𝐻(𝐹𝑣)) = (𝑢𝐺𝑣))
2928coeq1d 5872 . . . . . . . . 9 ((𝜑𝑢𝑅𝑣𝑅) → (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣)) = ((𝑢𝐺𝑣) ∘ (𝑢𝐺𝑣)))
30 catciso.r . . . . . . . . . . 11 𝑅 = (Base‘𝑋)
31 eqid 2737 . . . . . . . . . . 11 (Hom ‘𝑋) = (Hom ‘𝑋)
32 eqid 2737 . . . . . . . . . . 11 (Hom ‘𝑌) = (Hom ‘𝑌)
33 catcisolem.1 . . . . . . . . . . . 12 (𝜑𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
34333ad2ant1 1134 . . . . . . . . . . 11 ((𝜑𝑢𝑅𝑣𝑅) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
3530, 31, 32, 34, 7, 9ffthf1o 17966 . . . . . . . . . 10 ((𝜑𝑢𝑅𝑣𝑅) → (𝑢𝐺𝑣):(𝑢(Hom ‘𝑋)𝑣)–1-1-onto→((𝐹𝑢)(Hom ‘𝑌)(𝐹𝑣)))
36 f1ococnv1 6877 . . . . . . . . . 10 ((𝑢𝐺𝑣):(𝑢(Hom ‘𝑋)𝑣)–1-1-onto→((𝐹𝑢)(Hom ‘𝑌)(𝐹𝑣)) → ((𝑢𝐺𝑣) ∘ (𝑢𝐺𝑣)) = ( I ↾ (𝑢(Hom ‘𝑋)𝑣)))
3735, 36syl 17 . . . . . . . . 9 ((𝜑𝑢𝑅𝑣𝑅) → ((𝑢𝐺𝑣) ∘ (𝑢𝐺𝑣)) = ( I ↾ (𝑢(Hom ‘𝑋)𝑣)))
3829, 37eqtrd 2777 . . . . . . . 8 ((𝜑𝑢𝑅𝑣𝑅) → (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣)) = ( I ↾ (𝑢(Hom ‘𝑋)𝑣)))
3938mpoeq3dva 7510 . . . . . . 7 (𝜑 → (𝑢𝑅, 𝑣𝑅 ↦ (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣))) = (𝑢𝑅, 𝑣𝑅 ↦ ( I ↾ (𝑢(Hom ‘𝑋)𝑣))))
40 fveq2 6906 . . . . . . . . . 10 (𝑧 = ⟨𝑢, 𝑣⟩ → ((Hom ‘𝑋)‘𝑧) = ((Hom ‘𝑋)‘⟨𝑢, 𝑣⟩))
41 df-ov 7434 . . . . . . . . . 10 (𝑢(Hom ‘𝑋)𝑣) = ((Hom ‘𝑋)‘⟨𝑢, 𝑣⟩)
4240, 41eqtr4di 2795 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ → ((Hom ‘𝑋)‘𝑧) = (𝑢(Hom ‘𝑋)𝑣))
4342reseq2d 5997 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → ( I ↾ ((Hom ‘𝑋)‘𝑧)) = ( I ↾ (𝑢(Hom ‘𝑋)𝑣)))
4443mpompt 7547 . . . . . . 7 (𝑧 ∈ (𝑅 × 𝑅) ↦ ( I ↾ ((Hom ‘𝑋)‘𝑧))) = (𝑢𝑅, 𝑣𝑅 ↦ ( I ↾ (𝑢(Hom ‘𝑋)𝑣)))
4539, 44eqtr4di 2795 . . . . . 6 (𝜑 → (𝑢𝑅, 𝑣𝑅 ↦ (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣))) = (𝑧 ∈ (𝑅 × 𝑅) ↦ ( I ↾ ((Hom ‘𝑋)‘𝑧))))
463, 45opeq12d 4881 . . . . 5 (𝜑 → ⟨(𝐹𝐹), (𝑢𝑅, 𝑣𝑅 ↦ (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣)))⟩ = ⟨( I ↾ 𝑅), (𝑧 ∈ (𝑅 × 𝑅) ↦ ( I ↾ ((Hom ‘𝑋)‘𝑧)))⟩)
47 inss1 4237 . . . . . . . . 9 ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Full 𝑌)
48 fullfunc 17953 . . . . . . . . 9 (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
4947, 48sstri 3993 . . . . . . . 8 ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Func 𝑌)
5049ssbri 5188 . . . . . . 7 (𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺𝐹(𝑋 Func 𝑌)𝐺)
5133, 50syl 17 . . . . . 6 (𝜑𝐹(𝑋 Func 𝑌)𝐺)
52 catciso.s . . . . . . 7 𝑆 = (Base‘𝑌)
53 eqid 2737 . . . . . . 7 (Id‘𝑌) = (Id‘𝑌)
54 eqid 2737 . . . . . . 7 (Id‘𝑋) = (Id‘𝑋)
55 eqid 2737 . . . . . . 7 (comp‘𝑌) = (comp‘𝑌)
56 eqid 2737 . . . . . . 7 (comp‘𝑋) = (comp‘𝑋)
57 catciso.c . . . . . . . . . 10 𝐶 = (CatCat‘𝑈)
58 catciso.b . . . . . . . . . 10 𝐵 = (Base‘𝐶)
59 catciso.u . . . . . . . . . 10 (𝜑𝑈𝑉)
6057, 58, 59catcbas 18146 . . . . . . . . 9 (𝜑𝐵 = (𝑈 ∩ Cat))
61 inss2 4238 . . . . . . . . 9 (𝑈 ∩ Cat) ⊆ Cat
6260, 61eqsstrdi 4028 . . . . . . . 8 (𝜑𝐵 ⊆ Cat)
63 catciso.y . . . . . . . 8 (𝜑𝑌𝐵)
6462, 63sseldd 3984 . . . . . . 7 (𝜑𝑌 ∈ Cat)
65 catciso.x . . . . . . . 8 (𝜑𝑋𝐵)
6662, 65sseldd 3984 . . . . . . 7 (𝜑𝑋 ∈ Cat)
67 f1ocnv 6860 . . . . . . . 8 (𝐹:𝑅1-1-onto𝑆𝐹:𝑆1-1-onto𝑅)
68 f1of 6848 . . . . . . . 8 (𝐹:𝑆1-1-onto𝑅𝐹:𝑆𝑅)
691, 67, 683syl 18 . . . . . . 7 (𝜑𝐹:𝑆𝑅)
70 ovex 7464 . . . . . . . . . 10 ((𝐹𝑥)𝐺(𝐹𝑦)) ∈ V
7170cnvex 7947 . . . . . . . . 9 ((𝐹𝑥)𝐺(𝐹𝑦)) ∈ V
7217, 71fnmpoi 8095 . . . . . . . 8 𝐻 Fn (𝑆 × 𝑆)
7372a1i 11 . . . . . . 7 (𝜑𝐻 Fn (𝑆 × 𝑆))
7433adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
7569ffvelcdmda 7104 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → (𝐹𝑢) ∈ 𝑅)
7675adantrr 717 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝐹𝑢) ∈ 𝑅)
7769ffvelcdmda 7104 . . . . . . . . . . 11 ((𝜑𝑣𝑆) → (𝐹𝑣) ∈ 𝑅)
7877adantrl 716 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝐹𝑣) ∈ 𝑅)
7930, 31, 32, 74, 76, 78ffthf1o 17966 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))))
80 f1ocnv 6860 . . . . . . . . 9 (((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣)))–1-1-onto→((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
81 f1of 6848 . . . . . . . . 9 (((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣)))–1-1-onto→((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣)))⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
8279, 80, 813syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣)))⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
83 simpl 482 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥 = 𝑢)
8483fveq2d 6910 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝐹𝑥) = (𝐹𝑢))
85 simpr 484 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
8685fveq2d 6910 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝐹𝑦) = (𝐹𝑣))
8784, 86oveq12d 7449 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑣)))
8887cnveqd 5886 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑣)))
89 ovex 7464 . . . . . . . . . . . 12 ((𝐹𝑢)𝐺(𝐹𝑣)) ∈ V
9089cnvex 7947 . . . . . . . . . . 11 ((𝐹𝑢)𝐺(𝐹𝑣)) ∈ V
9188, 17, 90ovmpoa 7588 . . . . . . . . . 10 ((𝑢𝑆𝑣𝑆) → (𝑢𝐻𝑣) = ((𝐹𝑢)𝐺(𝐹𝑣)))
9291adantl 481 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝑢𝐻𝑣) = ((𝐹𝑢)𝐺(𝐹𝑣)))
931adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → 𝐹:𝑅1-1-onto𝑆)
94 simprl 771 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → 𝑢𝑆)
95 f1ocnvfv2 7297 . . . . . . . . . . . 12 ((𝐹:𝑅1-1-onto𝑆𝑢𝑆) → (𝐹‘(𝐹𝑢)) = 𝑢)
9693, 94, 95syl2anc 584 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝐹‘(𝐹𝑢)) = 𝑢)
97 simprr 773 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → 𝑣𝑆)
98 f1ocnvfv2 7297 . . . . . . . . . . . 12 ((𝐹:𝑅1-1-onto𝑆𝑣𝑆) → (𝐹‘(𝐹𝑣)) = 𝑣)
9993, 97, 98syl2anc 584 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝐹‘(𝐹𝑣)) = 𝑣)
10096, 99oveq12d 7449 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → ((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) = (𝑢(Hom ‘𝑌)𝑣))
101100eqcomd 2743 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝑢(Hom ‘𝑌)𝑣) = ((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))))
10292, 101feq12d 6724 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → ((𝑢𝐻𝑣):(𝑢(Hom ‘𝑌)𝑣)⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)) ↔ ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣)))⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))))
10382, 102mpbird 257 . . . . . . 7 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝑢𝐻𝑣):(𝑢(Hom ‘𝑌)𝑣)⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
104 simpr 484 . . . . . . . . . 10 ((𝜑𝑢𝑆) → 𝑢𝑆)
105 simpl 482 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑢) → 𝑥 = 𝑢)
106105fveq2d 6910 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑢) → (𝐹𝑥) = (𝐹𝑢))
107 simpr 484 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑢) → 𝑦 = 𝑢)
108107fveq2d 6910 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑢) → (𝐹𝑦) = (𝐹𝑢))
109106, 108oveq12d 7449 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑢) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑢)))
110109cnveqd 5886 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑢) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑢)))
111 ovex 7464 . . . . . . . . . . . 12 ((𝐹𝑢)𝐺(𝐹𝑢)) ∈ V
112111cnvex 7947 . . . . . . . . . . 11 ((𝐹𝑢)𝐺(𝐹𝑢)) ∈ V
113110, 17, 112ovmpoa 7588 . . . . . . . . . 10 ((𝑢𝑆𝑢𝑆) → (𝑢𝐻𝑢) = ((𝐹𝑢)𝐺(𝐹𝑢)))
114104, 104, 113syl2anc 584 . . . . . . . . 9 ((𝜑𝑢𝑆) → (𝑢𝐻𝑢) = ((𝐹𝑢)𝐺(𝐹𝑢)))
115114fveq1d 6908 . . . . . . . 8 ((𝜑𝑢𝑆) → ((𝑢𝐻𝑢)‘((Id‘𝑌)‘𝑢)) = (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑌)‘𝑢)))
11651adantr 480 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → 𝐹(𝑋 Func 𝑌)𝐺)
11730, 54, 53, 116, 75funcid 17915 . . . . . . . . . 10 ((𝜑𝑢𝑆) → (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑋)‘(𝐹𝑢))) = ((Id‘𝑌)‘(𝐹‘(𝐹𝑢))))
1181, 95sylan 580 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → (𝐹‘(𝐹𝑢)) = 𝑢)
119118fveq2d 6910 . . . . . . . . . 10 ((𝜑𝑢𝑆) → ((Id‘𝑌)‘(𝐹‘(𝐹𝑢))) = ((Id‘𝑌)‘𝑢))
120117, 119eqtrd 2777 . . . . . . . . 9 ((𝜑𝑢𝑆) → (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑋)‘(𝐹𝑢))) = ((Id‘𝑌)‘𝑢))
12133adantr 480 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
12230, 31, 32, 121, 75, 75ffthf1o 17966 . . . . . . . . . 10 ((𝜑𝑢𝑆) → ((𝐹𝑢)𝐺(𝐹𝑢)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑢))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑢))))
12366adantr 480 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → 𝑋 ∈ Cat)
12430, 31, 54, 123, 75catidcl 17725 . . . . . . . . . 10 ((𝜑𝑢𝑆) → ((Id‘𝑋)‘(𝐹𝑢)) ∈ ((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑢)))
125 f1ocnvfv 7298 . . . . . . . . . 10 ((((𝐹𝑢)𝐺(𝐹𝑢)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑢))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑢))) ∧ ((Id‘𝑋)‘(𝐹𝑢)) ∈ ((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑢))) → ((((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑋)‘(𝐹𝑢))) = ((Id‘𝑌)‘𝑢) → (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑌)‘𝑢)) = ((Id‘𝑋)‘(𝐹𝑢))))
126122, 124, 125syl2anc 584 . . . . . . . . 9 ((𝜑𝑢𝑆) → ((((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑋)‘(𝐹𝑢))) = ((Id‘𝑌)‘𝑢) → (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑌)‘𝑢)) = ((Id‘𝑋)‘(𝐹𝑢))))
127120, 126mpd 15 . . . . . . . 8 ((𝜑𝑢𝑆) → (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑌)‘𝑢)) = ((Id‘𝑋)‘(𝐹𝑢)))
128115, 127eqtrd 2777 . . . . . . 7 ((𝜑𝑢𝑆) → ((𝑢𝐻𝑢)‘((Id‘𝑌)‘𝑢)) = ((Id‘𝑋)‘(𝐹𝑢)))
129513ad2ant1 1134 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝐹(𝑋 Func 𝑌)𝐺)
130693ad2ant1 1134 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝐹:𝑆𝑅)
131 simp21 1207 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑢𝑆)
132130, 131ffvelcdmd 7105 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹𝑢) ∈ 𝑅)
133 simp22 1208 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑣𝑆)
134130, 133ffvelcdmd 7105 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹𝑣) ∈ 𝑅)
135 simp23 1209 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑧𝑆)
136130, 135ffvelcdmd 7105 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹𝑧) ∈ 𝑅)
137333ad2ant1 1134 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
13830, 31, 32, 137, 132, 134ffthf1o 17966 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))))
13913ad2ant1 1134 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝐹:𝑅1-1-onto𝑆)
140139, 131, 95syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹‘(𝐹𝑢)) = 𝑢)
141139, 133, 98syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹‘(𝐹𝑣)) = 𝑣)
142140, 141oveq12d 7449 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) = (𝑢(Hom ‘𝑌)𝑣))
143142f1oeq3d 6845 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) ↔ ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣)))
144138, 143mpbid 232 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣))
145 f1ocnv 6860 . . . . . . . . . . . . 13 (((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣) → ((𝐹𝑢)𝐺(𝐹𝑣)):(𝑢(Hom ‘𝑌)𝑣)–1-1-onto→((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
146 f1of 6848 . . . . . . . . . . . . 13 (((𝐹𝑢)𝐺(𝐹𝑣)):(𝑢(Hom ‘𝑌)𝑣)–1-1-onto→((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)) → ((𝐹𝑢)𝐺(𝐹𝑣)):(𝑢(Hom ‘𝑌)𝑣)⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
147144, 145, 1463syl 18 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑢)𝐺(𝐹𝑣)):(𝑢(Hom ‘𝑌)𝑣)⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
148 simp3l 1202 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣))
149147, 148ffvelcdmd 7105 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓) ∈ ((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
15030, 31, 32, 137, 134, 136ffthf1o 17966 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→((𝐹‘(𝐹𝑣))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))))
151 f1ocnvfv2 7297 . . . . . . . . . . . . . . . . 17 ((𝐹:𝑅1-1-onto𝑆𝑧𝑆) → (𝐹‘(𝐹𝑧)) = 𝑧)
152139, 135, 151syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹‘(𝐹𝑧)) = 𝑧)
153141, 152oveq12d 7449 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹‘(𝐹𝑣))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))) = (𝑣(Hom ‘𝑌)𝑧))
154153f1oeq3d 6845 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→((𝐹‘(𝐹𝑣))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))) ↔ ((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑣(Hom ‘𝑌)𝑧)))
155150, 154mpbid 232 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑣(Hom ‘𝑌)𝑧))
156 f1ocnv 6860 . . . . . . . . . . . . 13 (((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑣(Hom ‘𝑌)𝑧) → ((𝐹𝑣)𝐺(𝐹𝑧)):(𝑣(Hom ‘𝑌)𝑧)–1-1-onto→((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧)))
157 f1of 6848 . . . . . . . . . . . . 13 (((𝐹𝑣)𝐺(𝐹𝑧)):(𝑣(Hom ‘𝑌)𝑧)–1-1-onto→((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧)) → ((𝐹𝑣)𝐺(𝐹𝑧)):(𝑣(Hom ‘𝑌)𝑧)⟶((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧)))
158155, 156, 1573syl 18 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑣)𝐺(𝐹𝑧)):(𝑣(Hom ‘𝑌)𝑧)⟶((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧)))
159 simp3r 1203 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))
160158, 159ffvelcdmd 7105 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔) ∈ ((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧)))
16130, 31, 56, 55, 129, 132, 134, 136, 149, 160funcco 17916 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑧))‘((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))) = ((((𝐹𝑣)𝐺(𝐹𝑧))‘(((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔))(⟨(𝐹‘(𝐹𝑢)), (𝐹‘(𝐹𝑣))⟩(comp‘𝑌)(𝐹‘(𝐹𝑧)))(((𝐹𝑢)𝐺(𝐹𝑣))‘(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))))
162140, 141opeq12d 4881 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ⟨(𝐹‘(𝐹𝑢)), (𝐹‘(𝐹𝑣))⟩ = ⟨𝑢, 𝑣⟩)
163162, 152oveq12d 7449 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (⟨(𝐹‘(𝐹𝑢)), (𝐹‘(𝐹𝑣))⟩(comp‘𝑌)(𝐹‘(𝐹𝑧))) = (⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧))
164 f1ocnvfv2 7297 . . . . . . . . . . . 12 ((((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑣(Hom ‘𝑌)𝑧) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧)) → (((𝐹𝑣)𝐺(𝐹𝑧))‘(((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)) = 𝑔)
165155, 159, 164syl2anc 584 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑣)𝐺(𝐹𝑧))‘(((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)) = 𝑔)
166 f1ocnvfv2 7297 . . . . . . . . . . . 12 ((((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣) ∧ 𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣)) → (((𝐹𝑢)𝐺(𝐹𝑣))‘(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓)) = 𝑓)
167144, 148, 166syl2anc 584 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑣))‘(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓)) = 𝑓)
168163, 165, 167oveq123d 7452 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((((𝐹𝑣)𝐺(𝐹𝑧))‘(((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔))(⟨(𝐹‘(𝐹𝑢)), (𝐹‘(𝐹𝑣))⟩(comp‘𝑌)(𝐹‘(𝐹𝑧)))(((𝐹𝑢)𝐺(𝐹𝑣))‘(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))) = (𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓))
169161, 168eqtrd 2777 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑧))‘((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))) = (𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓))
17030, 31, 32, 137, 132, 136ffthf1o 17966 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑢)𝐺(𝐹𝑧)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))))
171140, 152oveq12d 7449 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))) = (𝑢(Hom ‘𝑌)𝑧))
172171f1oeq3d 6845 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑧)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))) ↔ ((𝐹𝑢)𝐺(𝐹𝑧)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑢(Hom ‘𝑌)𝑧)))
173170, 172mpbid 232 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑢)𝐺(𝐹𝑧)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑢(Hom ‘𝑌)𝑧))
174663ad2ant1 1134 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑋 ∈ Cat)
17530, 31, 56, 174, 132, 134, 136, 149, 160catcocl 17728 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓)) ∈ ((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧)))
176 f1ocnvfv 7298 . . . . . . . . . 10 ((((𝐹𝑢)𝐺(𝐹𝑧)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑢(Hom ‘𝑌)𝑧) ∧ ((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓)) ∈ ((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))) → ((((𝐹𝑢)𝐺(𝐹𝑧))‘((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))) = (𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓) → (((𝐹𝑢)𝐺(𝐹𝑧))‘(𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓)) = ((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))))
177173, 175, 176syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((((𝐹𝑢)𝐺(𝐹𝑧))‘((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))) = (𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓) → (((𝐹𝑢)𝐺(𝐹𝑧))‘(𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓)) = ((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))))
178169, 177mpd 15 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑧))‘(𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓)) = ((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓)))
179 simpl 482 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑧) → 𝑥 = 𝑢)
180179fveq2d 6910 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑧) → (𝐹𝑥) = (𝐹𝑢))
181 simpr 484 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑧) → 𝑦 = 𝑧)
182181fveq2d 6910 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑧) → (𝐹𝑦) = (𝐹𝑧))
183180, 182oveq12d 7449 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑧) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑧)))
184183cnveqd 5886 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑧) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑧)))
185 ovex 7464 . . . . . . . . . . . 12 ((𝐹𝑢)𝐺(𝐹𝑧)) ∈ V
186185cnvex 7947 . . . . . . . . . . 11 ((𝐹𝑢)𝐺(𝐹𝑧)) ∈ V
187184, 17, 186ovmpoa 7588 . . . . . . . . . 10 ((𝑢𝑆𝑧𝑆) → (𝑢𝐻𝑧) = ((𝐹𝑢)𝐺(𝐹𝑧)))
188131, 135, 187syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝑢𝐻𝑧) = ((𝐹𝑢)𝐺(𝐹𝑧)))
189188fveq1d 6908 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝑢𝐻𝑧)‘(𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓)) = (((𝐹𝑢)𝐺(𝐹𝑧))‘(𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓)))
190 simpl 482 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑣𝑦 = 𝑧) → 𝑥 = 𝑣)
191190fveq2d 6910 . . . . . . . . . . . . . 14 ((𝑥 = 𝑣𝑦 = 𝑧) → (𝐹𝑥) = (𝐹𝑣))
192 simpr 484 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑣𝑦 = 𝑧) → 𝑦 = 𝑧)
193192fveq2d 6910 . . . . . . . . . . . . . 14 ((𝑥 = 𝑣𝑦 = 𝑧) → (𝐹𝑦) = (𝐹𝑧))
194191, 193oveq12d 7449 . . . . . . . . . . . . 13 ((𝑥 = 𝑣𝑦 = 𝑧) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑣)𝐺(𝐹𝑧)))
195194cnveqd 5886 . . . . . . . . . . . 12 ((𝑥 = 𝑣𝑦 = 𝑧) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑣)𝐺(𝐹𝑧)))
196 ovex 7464 . . . . . . . . . . . . 13 ((𝐹𝑣)𝐺(𝐹𝑧)) ∈ V
197196cnvex 7947 . . . . . . . . . . . 12 ((𝐹𝑣)𝐺(𝐹𝑧)) ∈ V
198195, 17, 197ovmpoa 7588 . . . . . . . . . . 11 ((𝑣𝑆𝑧𝑆) → (𝑣𝐻𝑧) = ((𝐹𝑣)𝐺(𝐹𝑧)))
199133, 135, 198syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝑣𝐻𝑧) = ((𝐹𝑣)𝐺(𝐹𝑧)))
200199fveq1d 6908 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝑣𝐻𝑧)‘𝑔) = (((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔))
201131, 133, 91syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝑢𝐻𝑣) = ((𝐹𝑢)𝐺(𝐹𝑣)))
202201fveq1d 6908 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝑢𝐻𝑣)‘𝑓) = (((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))
203200, 202oveq12d 7449 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝑣𝐻𝑧)‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))((𝑢𝐻𝑣)‘𝑓)) = ((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓)))
204178, 189, 2033eqtr4d 2787 . . . . . . 7 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝑢𝐻𝑧)‘(𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓)) = (((𝑣𝐻𝑧)‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))((𝑢𝐻𝑣)‘𝑓)))
20552, 30, 32, 31, 53, 54, 55, 56, 64, 66, 69, 73, 103, 128, 204isfuncd 17910 . . . . . 6 (𝜑𝐹(𝑌 Func 𝑋)𝐻)
20630, 51, 205cofuval2 17932 . . . . 5 (𝜑 → (⟨𝐹, 𝐻⟩ ∘func𝐹, 𝐺⟩) = ⟨(𝐹𝐹), (𝑢𝑅, 𝑣𝑅 ↦ (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣)))⟩)
207 eqid 2737 . . . . . 6 (idfunc𝑋) = (idfunc𝑋)
208207, 30, 66, 31idfuval 17921 . . . . 5 (𝜑 → (idfunc𝑋) = ⟨( I ↾ 𝑅), (𝑧 ∈ (𝑅 × 𝑅) ↦ ( I ↾ ((Hom ‘𝑋)‘𝑧)))⟩)
20946, 206, 2083eqtr4d 2787 . . . 4 (𝜑 → (⟨𝐹, 𝐻⟩ ∘func𝐹, 𝐺⟩) = (idfunc𝑋))
210 eqid 2737 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
211 df-br 5144 . . . . . 6 (𝐹(𝑋 Func 𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑋 Func 𝑌))
21251, 211sylib 218 . . . . 5 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋 Func 𝑌))
213 df-br 5144 . . . . . 6 (𝐹(𝑌 Func 𝑋)𝐻 ↔ ⟨𝐹, 𝐻⟩ ∈ (𝑌 Func 𝑋))
214205, 213sylib 218 . . . . 5 (𝜑 → ⟨𝐹, 𝐻⟩ ∈ (𝑌 Func 𝑋))
21557, 58, 59, 210, 65, 63, 65, 212, 214catcco 18150 . . . 4 (𝜑 → (⟨𝐹, 𝐻⟩(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)⟨𝐹, 𝐺⟩) = (⟨𝐹, 𝐻⟩ ∘func𝐹, 𝐺⟩))
216 eqid 2737 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
21757, 58, 216, 207, 59, 65catcid 18152 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑋) = (idfunc𝑋))
218209, 215, 2173eqtr4d 2787 . . 3 (𝜑 → (⟨𝐹, 𝐻⟩(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)⟨𝐹, 𝐺⟩) = ((Id‘𝐶)‘𝑋))
219 eqid 2737 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
220 eqid 2737 . . . 4 (Sect‘𝐶) = (Sect‘𝐶)
22157catccat 18153 . . . . 5 (𝑈𝑉𝐶 ∈ Cat)
22259, 221syl 17 . . . 4 (𝜑𝐶 ∈ Cat)
22357, 58, 59, 219, 65, 63catchom 18148 . . . . 5 (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
224212, 223eleqtrrd 2844 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋(Hom ‘𝐶)𝑌))
22557, 58, 59, 219, 63, 65catchom 18148 . . . . 5 (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
226214, 225eleqtrrd 2844 . . . 4 (𝜑 → ⟨𝐹, 𝐻⟩ ∈ (𝑌(Hom ‘𝐶)𝑋))
22758, 219, 210, 216, 220, 222, 65, 63, 224, 226issect2 17798 . . 3 (𝜑 → (⟨𝐹, 𝐺⟩(𝑋(Sect‘𝐶)𝑌)⟨𝐹, 𝐻⟩ ↔ (⟨𝐹, 𝐻⟩(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)⟨𝐹, 𝐺⟩) = ((Id‘𝐶)‘𝑋)))
228218, 227mpbird 257 . 2 (𝜑 → ⟨𝐹, 𝐺⟩(𝑋(Sect‘𝐶)𝑌)⟨𝐹, 𝐻⟩)
229 f1ococnv2 6875 . . . . . . 7 (𝐹:𝑅1-1-onto𝑆 → (𝐹𝐹) = ( I ↾ 𝑆))
2301, 229syl 17 . . . . . 6 (𝜑 → (𝐹𝐹) = ( I ↾ 𝑆))
231913adant1 1131 . . . . . . . . . 10 ((𝜑𝑢𝑆𝑣𝑆) → (𝑢𝐻𝑣) = ((𝐹𝑢)𝐺(𝐹𝑣)))
232231coeq2d 5873 . . . . . . . . 9 ((𝜑𝑢𝑆𝑣𝑆) → (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣)) = (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ ((𝐹𝑢)𝐺(𝐹𝑣))))
233333ad2ant1 1134 . . . . . . . . . . . 12 ((𝜑𝑢𝑆𝑣𝑆) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
234753adant3 1133 . . . . . . . . . . . 12 ((𝜑𝑢𝑆𝑣𝑆) → (𝐹𝑢) ∈ 𝑅)
235773adant2 1132 . . . . . . . . . . . 12 ((𝜑𝑢𝑆𝑣𝑆) → (𝐹𝑣) ∈ 𝑅)
23630, 31, 32, 233, 234, 235ffthf1o 17966 . . . . . . . . . . 11 ((𝜑𝑢𝑆𝑣𝑆) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))))
2371003impb 1115 . . . . . . . . . . . 12 ((𝜑𝑢𝑆𝑣𝑆) → ((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) = (𝑢(Hom ‘𝑌)𝑣))
238237f1oeq3d 6845 . . . . . . . . . . 11 ((𝜑𝑢𝑆𝑣𝑆) → (((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) ↔ ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣)))
239236, 238mpbid 232 . . . . . . . . . 10 ((𝜑𝑢𝑆𝑣𝑆) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣))
240 f1ococnv2 6875 . . . . . . . . . 10 (((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣) → (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ ((𝐹𝑢)𝐺(𝐹𝑣))) = ( I ↾ (𝑢(Hom ‘𝑌)𝑣)))
241239, 240syl 17 . . . . . . . . 9 ((𝜑𝑢𝑆𝑣𝑆) → (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ ((𝐹𝑢)𝐺(𝐹𝑣))) = ( I ↾ (𝑢(Hom ‘𝑌)𝑣)))
242232, 241eqtrd 2777 . . . . . . . 8 ((𝜑𝑢𝑆𝑣𝑆) → (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣)) = ( I ↾ (𝑢(Hom ‘𝑌)𝑣)))
243242mpoeq3dva 7510 . . . . . . 7 (𝜑 → (𝑢𝑆, 𝑣𝑆 ↦ (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣))) = (𝑢𝑆, 𝑣𝑆 ↦ ( I ↾ (𝑢(Hom ‘𝑌)𝑣))))
244 fveq2 6906 . . . . . . . . . 10 (𝑧 = ⟨𝑢, 𝑣⟩ → ((Hom ‘𝑌)‘𝑧) = ((Hom ‘𝑌)‘⟨𝑢, 𝑣⟩))
245 df-ov 7434 . . . . . . . . . 10 (𝑢(Hom ‘𝑌)𝑣) = ((Hom ‘𝑌)‘⟨𝑢, 𝑣⟩)
246244, 245eqtr4di 2795 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ → ((Hom ‘𝑌)‘𝑧) = (𝑢(Hom ‘𝑌)𝑣))
247246reseq2d 5997 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → ( I ↾ ((Hom ‘𝑌)‘𝑧)) = ( I ↾ (𝑢(Hom ‘𝑌)𝑣)))
248247mpompt 7547 . . . . . . 7 (𝑧 ∈ (𝑆 × 𝑆) ↦ ( I ↾ ((Hom ‘𝑌)‘𝑧))) = (𝑢𝑆, 𝑣𝑆 ↦ ( I ↾ (𝑢(Hom ‘𝑌)𝑣)))
249243, 248eqtr4di 2795 . . . . . 6 (𝜑 → (𝑢𝑆, 𝑣𝑆 ↦ (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣))) = (𝑧 ∈ (𝑆 × 𝑆) ↦ ( I ↾ ((Hom ‘𝑌)‘𝑧))))
250230, 249opeq12d 4881 . . . . 5 (𝜑 → ⟨(𝐹𝐹), (𝑢𝑆, 𝑣𝑆 ↦ (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣)))⟩ = ⟨( I ↾ 𝑆), (𝑧 ∈ (𝑆 × 𝑆) ↦ ( I ↾ ((Hom ‘𝑌)‘𝑧)))⟩)
25152, 205, 51cofuval2 17932 . . . . 5 (𝜑 → (⟨𝐹, 𝐺⟩ ∘func𝐹, 𝐻⟩) = ⟨(𝐹𝐹), (𝑢𝑆, 𝑣𝑆 ↦ (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣)))⟩)
252 eqid 2737 . . . . . 6 (idfunc𝑌) = (idfunc𝑌)
253252, 52, 64, 32idfuval 17921 . . . . 5 (𝜑 → (idfunc𝑌) = ⟨( I ↾ 𝑆), (𝑧 ∈ (𝑆 × 𝑆) ↦ ( I ↾ ((Hom ‘𝑌)‘𝑧)))⟩)
254250, 251, 2533eqtr4d 2787 . . . 4 (𝜑 → (⟨𝐹, 𝐺⟩ ∘func𝐹, 𝐻⟩) = (idfunc𝑌))
25557, 58, 59, 210, 63, 65, 63, 214, 212catcco 18150 . . . 4 (𝜑 → (⟨𝐹, 𝐺⟩(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)⟨𝐹, 𝐻⟩) = (⟨𝐹, 𝐺⟩ ∘func𝐹, 𝐻⟩))
25657, 58, 216, 252, 59, 63catcid 18152 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑌) = (idfunc𝑌))
257254, 255, 2563eqtr4d 2787 . . 3 (𝜑 → (⟨𝐹, 𝐺⟩(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)⟨𝐹, 𝐻⟩) = ((Id‘𝐶)‘𝑌))
25858, 219, 210, 216, 220, 222, 63, 65, 226, 224issect2 17798 . . 3 (𝜑 → (⟨𝐹, 𝐻⟩(𝑌(Sect‘𝐶)𝑋)⟨𝐹, 𝐺⟩ ↔ (⟨𝐹, 𝐺⟩(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)⟨𝐹, 𝐻⟩) = ((Id‘𝐶)‘𝑌)))
259257, 258mpbird 257 . 2 (𝜑 → ⟨𝐹, 𝐻⟩(𝑌(Sect‘𝐶)𝑋)⟨𝐹, 𝐺⟩)
260 catcisolem.i . . 3 𝐼 = (Inv‘𝐶)
26158, 260, 222, 65, 63, 220isinv 17804 . 2 (𝜑 → (⟨𝐹, 𝐺⟩(𝑋𝐼𝑌)⟨𝐹, 𝐻⟩ ↔ (⟨𝐹, 𝐺⟩(𝑋(Sect‘𝐶)𝑌)⟨𝐹, 𝐻⟩ ∧ ⟨𝐹, 𝐻⟩(𝑌(Sect‘𝐶)𝑋)⟨𝐹, 𝐺⟩)))
262228, 259, 261mpbir2and 713 1 (𝜑 → ⟨𝐹, 𝐺⟩(𝑋𝐼𝑌)⟨𝐹, 𝐻⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  cin 3950  cop 4632   class class class wbr 5143  cmpt 5225   I cid 5577   × cxp 5683  ccnv 5684  cres 5687  ccom 5689   Fn wfn 6556  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Idccid 17708  Sectcsect 17788  Invcinv 17789   Func cfunc 17899  idfunccidfu 17900  func ccofu 17901   Full cful 17949   Faith cfth 17950  CatCatccatc 18143
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-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  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-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-cat 17711  df-cid 17712  df-sect 17791  df-inv 17792  df-func 17903  df-idfu 17904  df-cofu 17905  df-full 17951  df-fth 17952  df-catc 18144
This theorem is referenced by:  catciso  18156
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