| Step | Hyp | Ref
| Expression |
| 1 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑏 = 𝑎 → (𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏) = (𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)) |
| 2 | 1 | oveq1d 7446 |
. . . . . . 7
⊢ (𝑏 = 𝑎 → ((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾)) = ((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))) |
| 3 | 2 | fveq1d 6908 |
. . . . . 6
⊢ (𝑏 = 𝑎 → (((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃) = (((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃)) |
| 4 | 3 | fveq1d 6908 |
. . . . 5
⊢ (𝑏 = 𝑎 → ((((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃)‘( 1 ‘𝑃)) = ((((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃)‘( 1 ‘𝑃))) |
| 5 | 4 | cbvmptv 5255 |
. . . 4
⊢ (𝑏 ∈ (((1st
‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃)‘( 1 ‘𝑃))) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃)‘( 1 ‘𝑃))) |
| 6 | | yoneda.q |
. . . . . . . . 9
⊢ 𝑄 = (𝑂 FuncCat 𝑆) |
| 7 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆) |
| 8 | | yoneda.o |
. . . . . . . . . 10
⊢ 𝑂 = (oppCat‘𝐶) |
| 9 | | yoneda.b |
. . . . . . . . . 10
⊢ 𝐵 = (Base‘𝐶) |
| 10 | 8, 9 | oppcbas 17761 |
. . . . . . . . 9
⊢ 𝐵 = (Base‘𝑂) |
| 11 | | eqid 2737 |
. . . . . . . . 9
⊢
(comp‘𝑆) =
(comp‘𝑆) |
| 12 | | eqid 2737 |
. . . . . . . . 9
⊢
(comp‘𝑄) =
(comp‘𝑄) |
| 13 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
| 14 | 6, 7 | fuchom 18009 |
. . . . . . . . . . . 12
⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄) |
| 15 | | relfunc 17907 |
. . . . . . . . . . . . 13
⊢ Rel
(𝐶 Func 𝑄) |
| 16 | | yoneda.y |
. . . . . . . . . . . . . 14
⊢ 𝑌 = (Yon‘𝐶) |
| 17 | | yoneda.c |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐶 ∈ Cat) |
| 18 | | yoneda.s |
. . . . . . . . . . . . . 14
⊢ 𝑆 = (SetCat‘𝑈) |
| 19 | | yoneda.w |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑉 ∈ 𝑊) |
| 20 | | yoneda.v |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (ran
(Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉) |
| 21 | 20 | unssbd 4194 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
| 22 | 19, 21 | ssexd 5324 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑈 ∈ V) |
| 23 | | yoneda.u |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ran
(Homf ‘𝐶) ⊆ 𝑈) |
| 24 | 16, 17, 8, 18, 6, 22, 23 | yoncl 18307 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄)) |
| 25 | | 1st2ndbr 8067 |
. . . . . . . . . . . . 13
⊢ ((Rel
(𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) |
| 26 | 15, 24, 25 | sylancr 587 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st
‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) |
| 27 | | yonedalem22.p |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑃 ∈ 𝐵) |
| 28 | | yonedalem21.x |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| 29 | 9, 13, 14, 26, 27, 28 | funcf2 17913 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑃(2nd ‘𝑌)𝑋):(𝑃(Hom ‘𝐶)𝑋)⟶(((1st ‘𝑌)‘𝑃)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑋))) |
| 30 | | yonedalem22.k |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋)) |
| 31 | 29, 30 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑃(2nd ‘𝑌)𝑋)‘𝐾) ∈ (((1st ‘𝑌)‘𝑃)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑋))) |
| 32 | 31 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑃(2nd ‘𝑌)𝑋)‘𝐾) ∈ (((1st ‘𝑌)‘𝑃)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑋))) |
| 33 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) |
| 34 | | yonedalem22.a |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺)) |
| 35 | 34 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺)) |
| 36 | 6, 7, 12, 33, 35 | fuccocl 18012 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎) ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐺)) |
| 37 | 27 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑃 ∈ 𝐵) |
| 38 | 6, 7, 10, 11, 12, 32, 36, 37 | fuccoval 18011 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃) = (((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)‘𝑃)(〈((1st
‘((1st ‘𝑌)‘𝑃))‘𝑃), ((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)〉(comp‘𝑆)((1st ‘𝐺)‘𝑃))(((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃))) |
| 39 | 6, 7, 10, 11, 12, 33, 35, 37 | fuccoval 18011 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)‘𝑃) = ((𝐴‘𝑃)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃), ((1st ‘𝐹)‘𝑃)〉(comp‘𝑆)((1st ‘𝐺)‘𝑃))(𝑎‘𝑃))) |
| 40 | 22 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V) |
| 41 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(Base‘𝑆) =
(Base‘𝑆) |
| 42 | | relfunc 17907 |
. . . . . . . . . . . . . . . 16
⊢ Rel
(𝑂 Func 𝑆) |
| 43 | 6 | fucbas 18008 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑂 Func 𝑆) = (Base‘𝑄) |
| 44 | 9, 43, 26 | funcf1 17911 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (1st
‘𝑌):𝐵⟶(𝑂 Func 𝑆)) |
| 45 | 44, 28 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((1st
‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) |
| 46 | | 1st2ndbr 8067 |
. . . . . . . . . . . . . . . 16
⊢ ((Rel
(𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1st
‘((1st ‘𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑋))) |
| 47 | 42, 45, 46 | sylancr 587 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (1st
‘((1st ‘𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑋))) |
| 48 | 10, 41, 47 | funcf1 17911 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1st
‘((1st ‘𝑌)‘𝑋)):𝐵⟶(Base‘𝑆)) |
| 49 | 18, 22 | setcbas 18123 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑈 = (Base‘𝑆)) |
| 50 | 49 | feq3d 6723 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((1st
‘((1st ‘𝑌)‘𝑋)):𝐵⟶𝑈 ↔ (1st
‘((1st ‘𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))) |
| 51 | 48, 50 | mpbird 257 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1st
‘((1st ‘𝑌)‘𝑋)):𝐵⟶𝑈) |
| 52 | 51, 27 | ffvelcdmd 7105 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃) ∈ 𝑈) |
| 53 | 52 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃) ∈ 𝑈) |
| 54 | | yonedalem21.f |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆)) |
| 55 | | 1st2ndbr 8067 |
. . . . . . . . . . . . . . . 16
⊢ ((Rel
(𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st ‘𝐹)(𝑂 Func 𝑆)(2nd ‘𝐹)) |
| 56 | 42, 54, 55 | sylancr 587 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (1st
‘𝐹)(𝑂 Func 𝑆)(2nd ‘𝐹)) |
| 57 | 10, 41, 56 | funcf1 17911 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1st
‘𝐹):𝐵⟶(Base‘𝑆)) |
| 58 | 49 | feq3d 6723 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((1st
‘𝐹):𝐵⟶𝑈 ↔ (1st ‘𝐹):𝐵⟶(Base‘𝑆))) |
| 59 | 57, 58 | mpbird 257 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1st
‘𝐹):𝐵⟶𝑈) |
| 60 | 59, 27 | ffvelcdmd 7105 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((1st
‘𝐹)‘𝑃) ∈ 𝑈) |
| 61 | 60 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘𝐹)‘𝑃) ∈ 𝑈) |
| 62 | | yonedalem22.g |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐺 ∈ (𝑂 Func 𝑆)) |
| 63 | | 1st2ndbr 8067 |
. . . . . . . . . . . . . . . 16
⊢ ((Rel
(𝑂 Func 𝑆) ∧ 𝐺 ∈ (𝑂 Func 𝑆)) → (1st ‘𝐺)(𝑂 Func 𝑆)(2nd ‘𝐺)) |
| 64 | 42, 62, 63 | sylancr 587 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (1st
‘𝐺)(𝑂 Func 𝑆)(2nd ‘𝐺)) |
| 65 | 10, 41, 64 | funcf1 17911 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1st
‘𝐺):𝐵⟶(Base‘𝑆)) |
| 66 | 65, 27 | ffvelcdmd 7105 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((1st
‘𝐺)‘𝑃) ∈ (Base‘𝑆)) |
| 67 | 66, 49 | eleqtrrd 2844 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((1st
‘𝐺)‘𝑃) ∈ 𝑈) |
| 68 | 67 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘𝐺)‘𝑃) ∈ 𝑈) |
| 69 | 7, 33 | nat1st2nd 17999 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (〈(1st
‘((1st ‘𝑌)‘𝑋)), (2nd ‘((1st
‘𝑌)‘𝑋))〉(𝑂 Nat 𝑆)〈(1st ‘𝐹), (2nd ‘𝐹)〉)) |
| 70 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (Hom
‘𝑆) = (Hom
‘𝑆) |
| 71 | 7, 69, 10, 70, 37 | natcl 18001 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎‘𝑃) ∈ (((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃)(Hom ‘𝑆)((1st ‘𝐹)‘𝑃))) |
| 72 | 18, 40, 70, 53, 61 | elsetchom 18126 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑃) ∈ (((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃)(Hom ‘𝑆)((1st ‘𝐹)‘𝑃)) ↔ (𝑎‘𝑃):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)⟶((1st ‘𝐹)‘𝑃))) |
| 73 | 71, 72 | mpbid 232 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎‘𝑃):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)⟶((1st ‘𝐹)‘𝑃)) |
| 74 | 7, 34 | nat1st2nd 17999 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐴 ∈ (〈(1st ‘𝐹), (2nd ‘𝐹)〉(𝑂 Nat 𝑆)〈(1st ‘𝐺), (2nd ‘𝐺)〉)) |
| 75 | 7, 74, 10, 70, 27 | natcl 18001 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴‘𝑃) ∈ (((1st ‘𝐹)‘𝑃)(Hom ‘𝑆)((1st ‘𝐺)‘𝑃))) |
| 76 | 18, 22, 70, 60, 67 | elsetchom 18126 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐴‘𝑃) ∈ (((1st ‘𝐹)‘𝑃)(Hom ‘𝑆)((1st ‘𝐺)‘𝑃)) ↔ (𝐴‘𝑃):((1st ‘𝐹)‘𝑃)⟶((1st ‘𝐺)‘𝑃))) |
| 77 | 75, 76 | mpbid 232 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴‘𝑃):((1st ‘𝐹)‘𝑃)⟶((1st ‘𝐺)‘𝑃)) |
| 78 | 77 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐴‘𝑃):((1st ‘𝐹)‘𝑃)⟶((1st ‘𝐺)‘𝑃)) |
| 79 | 18, 40, 11, 53, 61, 68, 73, 78 | setcco 18128 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴‘𝑃)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃), ((1st ‘𝐹)‘𝑃)〉(comp‘𝑆)((1st ‘𝐺)‘𝑃))(𝑎‘𝑃)) = ((𝐴‘𝑃) ∘ (𝑎‘𝑃))) |
| 80 | 39, 79 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)‘𝑃) = ((𝐴‘𝑃) ∘ (𝑎‘𝑃))) |
| 81 | 80 | oveq1d 7446 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)‘𝑃)(〈((1st
‘((1st ‘𝑌)‘𝑃))‘𝑃), ((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)〉(comp‘𝑆)((1st ‘𝐺)‘𝑃))(((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)) = (((𝐴‘𝑃) ∘ (𝑎‘𝑃))(〈((1st
‘((1st ‘𝑌)‘𝑃))‘𝑃), ((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)〉(comp‘𝑆)((1st ‘𝐺)‘𝑃))(((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃))) |
| 82 | 44, 27 | ffvelcdmd 7105 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((1st
‘𝑌)‘𝑃) ∈ (𝑂 Func 𝑆)) |
| 83 | | 1st2ndbr 8067 |
. . . . . . . . . . . . . 14
⊢ ((Rel
(𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑃) ∈ (𝑂 Func 𝑆)) → (1st
‘((1st ‘𝑌)‘𝑃))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑃))) |
| 84 | 42, 82, 83 | sylancr 587 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1st
‘((1st ‘𝑌)‘𝑃))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑃))) |
| 85 | 10, 41, 84 | funcf1 17911 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st
‘((1st ‘𝑌)‘𝑃)):𝐵⟶(Base‘𝑆)) |
| 86 | 85, 27 | ffvelcdmd 7105 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1st
‘((1st ‘𝑌)‘𝑃))‘𝑃) ∈ (Base‘𝑆)) |
| 87 | 86, 49 | eleqtrrd 2844 |
. . . . . . . . . 10
⊢ (𝜑 → ((1st
‘((1st ‘𝑌)‘𝑃))‘𝑃) ∈ 𝑈) |
| 88 | 87 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st
‘((1st ‘𝑌)‘𝑃))‘𝑃) ∈ 𝑈) |
| 89 | 7, 31 | nat1st2nd 17999 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑃(2nd ‘𝑌)𝑋)‘𝐾) ∈ (〈(1st
‘((1st ‘𝑌)‘𝑃)), (2nd ‘((1st
‘𝑌)‘𝑃))〉(𝑂 Nat 𝑆)〈(1st
‘((1st ‘𝑌)‘𝑋)), (2nd ‘((1st
‘𝑌)‘𝑋))〉)) |
| 90 | 7, 89, 10, 70, 27 | natcl 18001 |
. . . . . . . . . . 11
⊢ (𝜑 → (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃) ∈ (((1st
‘((1st ‘𝑌)‘𝑃))‘𝑃)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃))) |
| 91 | 18, 22, 70, 87, 52 | elsetchom 18126 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃) ∈ (((1st
‘((1st ‘𝑌)‘𝑃))‘𝑃)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)) ↔ (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st
‘𝑌)‘𝑃))‘𝑃)⟶((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃))) |
| 92 | 90, 91 | mpbid 232 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st
‘𝑌)‘𝑃))‘𝑃)⟶((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃)) |
| 93 | 92 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st
‘𝑌)‘𝑃))‘𝑃)⟶((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃)) |
| 94 | | fco 6760 |
. . . . . . . . . 10
⊢ (((𝐴‘𝑃):((1st ‘𝐹)‘𝑃)⟶((1st ‘𝐺)‘𝑃) ∧ (𝑎‘𝑃):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)⟶((1st ‘𝐹)‘𝑃)) → ((𝐴‘𝑃) ∘ (𝑎‘𝑃)):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)⟶((1st ‘𝐺)‘𝑃)) |
| 95 | 78, 73, 94 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴‘𝑃) ∘ (𝑎‘𝑃)):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)⟶((1st ‘𝐺)‘𝑃)) |
| 96 | 18, 40, 11, 88, 53, 68, 93, 95 | setcco 18128 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝐴‘𝑃) ∘ (𝑎‘𝑃))(〈((1st
‘((1st ‘𝑌)‘𝑃))‘𝑃), ((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)〉(comp‘𝑆)((1st ‘𝐺)‘𝑃))(((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)) = (((𝐴‘𝑃) ∘ (𝑎‘𝑃)) ∘ (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃))) |
| 97 | 38, 81, 96 | 3eqtrd 2781 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃) = (((𝐴‘𝑃) ∘ (𝑎‘𝑃)) ∘ (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃))) |
| 98 | 97 | fveq1d 6908 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃)‘( 1 ‘𝑃)) = ((((𝐴‘𝑃) ∘ (𝑎‘𝑃)) ∘ (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃))‘( 1 ‘𝑃))) |
| 99 | | yoneda.1 |
. . . . . . . . . 10
⊢ 1 =
(Id‘𝐶) |
| 100 | 9, 13, 99, 17, 27 | catidcl 17725 |
. . . . . . . . 9
⊢ (𝜑 → ( 1 ‘𝑃) ∈ (𝑃(Hom ‘𝐶)𝑃)) |
| 101 | 16, 9, 17, 27, 13, 27 | yon11 18309 |
. . . . . . . . 9
⊢ (𝜑 → ((1st
‘((1st ‘𝑌)‘𝑃))‘𝑃) = (𝑃(Hom ‘𝐶)𝑃)) |
| 102 | 100, 101 | eleqtrrd 2844 |
. . . . . . . 8
⊢ (𝜑 → ( 1 ‘𝑃) ∈ ((1st
‘((1st ‘𝑌)‘𝑃))‘𝑃)) |
| 103 | 102 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1 ‘𝑃) ∈ ((1st
‘((1st ‘𝑌)‘𝑃))‘𝑃)) |
| 104 | | fvco3 7008 |
. . . . . . 7
⊢
(((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st
‘𝑌)‘𝑃))‘𝑃)⟶((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃) ∧ ( 1 ‘𝑃) ∈ ((1st
‘((1st ‘𝑌)‘𝑃))‘𝑃)) → ((((𝐴‘𝑃) ∘ (𝑎‘𝑃)) ∘ (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃))‘( 1 ‘𝑃)) = (((𝐴‘𝑃) ∘ (𝑎‘𝑃))‘((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃)))) |
| 105 | 93, 103, 104 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝐴‘𝑃) ∘ (𝑎‘𝑃)) ∘ (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃))‘( 1 ‘𝑃)) = (((𝐴‘𝑃) ∘ (𝑎‘𝑃))‘((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃)))) |
| 106 | 93, 103 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃)) ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃)) |
| 107 | | fvco3 7008 |
. . . . . . . 8
⊢ (((𝑎‘𝑃):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)⟶((1st ‘𝐹)‘𝑃) ∧ ((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃)) ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃)) → (((𝐴‘𝑃) ∘ (𝑎‘𝑃))‘((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃))) = ((𝐴‘𝑃)‘((𝑎‘𝑃)‘((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃))))) |
| 108 | 73, 106, 107 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝐴‘𝑃) ∘ (𝑎‘𝑃))‘((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃))) = ((𝐴‘𝑃)‘((𝑎‘𝑃)‘((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃))))) |
| 109 | 17 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐶 ∈ Cat) |
| 110 | 28 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑋 ∈ 𝐵) |
| 111 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(comp‘𝐶) =
(comp‘𝐶) |
| 112 | 30 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋)) |
| 113 | 100 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1 ‘𝑃) ∈ (𝑃(Hom ‘𝐶)𝑃)) |
| 114 | 16, 9, 109, 37, 13, 110, 111, 37, 112, 113 | yon2 18311 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃)) = (𝐾(〈𝑃, 𝑃〉(comp‘𝐶)𝑋)( 1 ‘𝑃))) |
| 115 | 9, 13, 99, 109, 37, 111, 110, 112 | catrid 17727 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐾(〈𝑃, 𝑃〉(comp‘𝐶)𝑋)( 1 ‘𝑃)) = 𝐾) |
| 116 | 114, 115 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃)) = 𝐾) |
| 117 | 116 | fveq2d 6910 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑃)‘((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃))) = ((𝑎‘𝑃)‘𝐾)) |
| 118 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢ (Hom
‘𝑂) = (Hom
‘𝑂) |
| 119 | 10, 118, 70, 47, 28, 27 | funcf2 17913 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃):(𝑋(Hom ‘𝑂)𝑃)⟶(((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃))) |
| 120 | 13, 8 | oppchom 17758 |
. . . . . . . . . . . . . . 15
⊢ (𝑋(Hom ‘𝑂)𝑃) = (𝑃(Hom ‘𝐶)𝑋) |
| 121 | 30, 120 | eleqtrrdi 2852 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃)) |
| 122 | 119, 121 | ffvelcdmd 7105 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾) ∈ (((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃))) |
| 123 | 51, 28 | ffvelcdmd 7105 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋) ∈ 𝑈) |
| 124 | 18, 22, 70, 123, 52 | elsetchom 18126 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾) ∈ (((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)) ↔ ((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑋)⟶((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃))) |
| 125 | 122, 124 | mpbid 232 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑋)⟶((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃)) |
| 126 | 125 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑋)⟶((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃)) |
| 127 | 9, 13, 99, 17, 28 | catidcl 17725 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
| 128 | 16, 9, 17, 28, 13, 28 | yon11 18309 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋)) |
| 129 | 127, 128 | eleqtrrd 2844 |
. . . . . . . . . . . 12
⊢ (𝜑 → ( 1 ‘𝑋) ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋)) |
| 130 | 129 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1 ‘𝑋) ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋)) |
| 131 | | fvco3 7008 |
. . . . . . . . . . 11
⊢ ((((𝑋(2nd
‘((1st ‘𝑌)‘𝑋))𝑃)‘𝐾):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑋)⟶((1st
‘((1st ‘𝑌)‘𝑋))‘𝑃) ∧ ( 1 ‘𝑋) ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋)) → (((𝑎‘𝑃) ∘ ((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾))‘( 1 ‘𝑋)) = ((𝑎‘𝑃)‘(((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾)‘( 1 ‘𝑋)))) |
| 132 | 126, 130,
131 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑎‘𝑃) ∘ ((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾))‘( 1 ‘𝑋)) = ((𝑎‘𝑃)‘(((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾)‘( 1 ‘𝑋)))) |
| 133 | 121 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃)) |
| 134 | 7, 69, 10, 118, 11, 110, 37, 133 | nati 18003 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑃)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋), ((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)〉(comp‘𝑆)((1st ‘𝐹)‘𝑃))((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾)) = (((𝑋(2nd ‘𝐹)𝑃)‘𝐾)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋), ((1st ‘𝐹)‘𝑋)〉(comp‘𝑆)((1st ‘𝐹)‘𝑃))(𝑎‘𝑋))) |
| 135 | 123 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋) ∈ 𝑈) |
| 136 | 18, 40, 11, 135, 53, 61, 126, 73 | setcco 18128 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑃)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋), ((1st ‘((1st
‘𝑌)‘𝑋))‘𝑃)〉(comp‘𝑆)((1st ‘𝐹)‘𝑃))((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾)) = ((𝑎‘𝑃) ∘ ((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾))) |
| 137 | 59, 28 | ffvelcdmd 7105 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((1st
‘𝐹)‘𝑋) ∈ 𝑈) |
| 138 | 137 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘𝐹)‘𝑋) ∈ 𝑈) |
| 139 | 7, 69, 10, 70, 110 | natcl 18001 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎‘𝑋) ∈ (((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑋))) |
| 140 | 18, 40, 70, 135, 138 | elsetchom 18126 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑋) ∈ (((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑋)) ↔ (𝑎‘𝑋):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑋)⟶((1st ‘𝐹)‘𝑋))) |
| 141 | 139, 140 | mpbid 232 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎‘𝑋):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑋)⟶((1st ‘𝐹)‘𝑋)) |
| 142 | 10, 118, 70, 56, 28, 27 | funcf2 17913 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑋(2nd ‘𝐹)𝑃):(𝑋(Hom ‘𝑂)𝑃)⟶(((1st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑃))) |
| 143 | 142, 121 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑋(2nd ‘𝐹)𝑃)‘𝐾) ∈ (((1st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑃))) |
| 144 | 18, 22, 70, 137, 60 | elsetchom 18126 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((𝑋(2nd ‘𝐹)𝑃)‘𝐾) ∈ (((1st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑃)) ↔ ((𝑋(2nd ‘𝐹)𝑃)‘𝐾):((1st ‘𝐹)‘𝑋)⟶((1st ‘𝐹)‘𝑃))) |
| 145 | 143, 144 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑋(2nd ‘𝐹)𝑃)‘𝐾):((1st ‘𝐹)‘𝑋)⟶((1st ‘𝐹)‘𝑃)) |
| 146 | 145 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑋(2nd ‘𝐹)𝑃)‘𝐾):((1st ‘𝐹)‘𝑋)⟶((1st ‘𝐹)‘𝑃)) |
| 147 | 18, 40, 11, 135, 138, 61, 141, 146 | setcco 18128 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑋(2nd ‘𝐹)𝑃)‘𝐾)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋), ((1st ‘𝐹)‘𝑋)〉(comp‘𝑆)((1st ‘𝐹)‘𝑃))(𝑎‘𝑋)) = (((𝑋(2nd ‘𝐹)𝑃)‘𝐾) ∘ (𝑎‘𝑋))) |
| 148 | 134, 136,
147 | 3eqtr3d 2785 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑃) ∘ ((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾)) = (((𝑋(2nd ‘𝐹)𝑃)‘𝐾) ∘ (𝑎‘𝑋))) |
| 149 | 148 | fveq1d 6908 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑎‘𝑃) ∘ ((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾))‘( 1 ‘𝑋)) = ((((𝑋(2nd ‘𝐹)𝑃)‘𝐾) ∘ (𝑎‘𝑋))‘( 1 ‘𝑋))) |
| 150 | 127 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
| 151 | 16, 9, 109, 110, 13, 110, 111, 37, 112, 150 | yon12 18310 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾)‘( 1 ‘𝑋)) = (( 1 ‘𝑋)(〈𝑃, 𝑋〉(comp‘𝐶)𝑋)𝐾)) |
| 152 | 9, 13, 99, 109, 37, 111, 110, 112 | catlid 17726 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (( 1 ‘𝑋)(〈𝑃, 𝑋〉(comp‘𝐶)𝑋)𝐾) = 𝐾) |
| 153 | 151, 152 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾)‘( 1 ‘𝑋)) = 𝐾) |
| 154 | 153 | fveq2d 6910 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑃)‘(((𝑋(2nd ‘((1st
‘𝑌)‘𝑋))𝑃)‘𝐾)‘( 1 ‘𝑋))) = ((𝑎‘𝑃)‘𝐾)) |
| 155 | 132, 149,
154 | 3eqtr3d 2785 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑋(2nd ‘𝐹)𝑃)‘𝐾) ∘ (𝑎‘𝑋))‘( 1 ‘𝑋)) = ((𝑎‘𝑃)‘𝐾)) |
| 156 | | fvco3 7008 |
. . . . . . . . . 10
⊢ (((𝑎‘𝑋):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑋)⟶((1st ‘𝐹)‘𝑋) ∧ ( 1 ‘𝑋) ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑋)) → ((((𝑋(2nd ‘𝐹)𝑃)‘𝐾) ∘ (𝑎‘𝑋))‘( 1 ‘𝑋)) = (((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘((𝑎‘𝑋)‘( 1 ‘𝑋)))) |
| 157 | 141, 130,
156 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑋(2nd ‘𝐹)𝑃)‘𝐾) ∘ (𝑎‘𝑋))‘( 1 ‘𝑋)) = (((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘((𝑎‘𝑋)‘( 1 ‘𝑋)))) |
| 158 | 117, 155,
157 | 3eqtr2d 2783 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑃)‘((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃))) = (((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘((𝑎‘𝑋)‘( 1 ‘𝑋)))) |
| 159 | 158 | fveq2d 6910 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴‘𝑃)‘((𝑎‘𝑃)‘((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃)))) = ((𝐴‘𝑃)‘(((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘((𝑎‘𝑋)‘( 1 ‘𝑋))))) |
| 160 | 108, 159 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝐴‘𝑃) ∘ (𝑎‘𝑃))‘((((𝑃(2nd ‘𝑌)𝑋)‘𝐾)‘𝑃)‘( 1 ‘𝑃))) = ((𝐴‘𝑃)‘(((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘((𝑎‘𝑋)‘( 1 ‘𝑋))))) |
| 161 | 98, 105, 160 | 3eqtrd 2781 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃)‘( 1 ‘𝑃)) = ((𝐴‘𝑃)‘(((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘((𝑎‘𝑋)‘( 1 ‘𝑋))))) |
| 162 | 161 | mpteq2dva 5242 |
. . . 4
⊢ (𝜑 → (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑎)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃)‘( 1 ‘𝑃))) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴‘𝑃)‘(((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘((𝑎‘𝑋)‘( 1 ‘𝑋)))))) |
| 163 | 5, 162 | eqtrid 2789 |
. . 3
⊢ (𝜑 → (𝑏 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃)‘( 1 ‘𝑃))) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴‘𝑃)‘(((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘((𝑎‘𝑋)‘( 1 ‘𝑋)))))) |
| 164 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑄 ×c
𝑂) = (𝑄 ×c 𝑂) |
| 165 | 164, 43, 10 | xpcbas 18223 |
. . . . . . . . . 10
⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂)) |
| 166 | | eqid 2737 |
. . . . . . . . . 10
⊢ (Hom
‘(𝑄
×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂)) |
| 167 | | eqid 2737 |
. . . . . . . . . 10
⊢ (Hom
‘𝑇) = (Hom
‘𝑇) |
| 168 | | relfunc 17907 |
. . . . . . . . . . 11
⊢ Rel
((𝑄
×c 𝑂) Func 𝑇) |
| 169 | | yoneda.t |
. . . . . . . . . . . . 13
⊢ 𝑇 = (SetCat‘𝑉) |
| 170 | | yoneda.h |
. . . . . . . . . . . . 13
⊢ 𝐻 =
(HomF‘𝑄) |
| 171 | | yoneda.r |
. . . . . . . . . . . . 13
⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇) |
| 172 | | yoneda.e |
. . . . . . . . . . . . 13
⊢ 𝐸 = (𝑂 evalF 𝑆) |
| 173 | | yoneda.z |
. . . . . . . . . . . . 13
⊢ 𝑍 = (𝐻 ∘func
((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉
∘func (𝑄 2ndF 𝑂))
〈,〉F (𝑄 1stF 𝑂))) |
| 174 | 16, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20 | yonedalem1 18317 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))) |
| 175 | 174 | simpld 494 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
| 176 | | 1st2ndbr 8067 |
. . . . . . . . . . 11
⊢ ((Rel
((𝑄
×c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st ‘𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd ‘𝑍)) |
| 177 | 168, 175,
176 | sylancr 587 |
. . . . . . . . . 10
⊢ (𝜑 → (1st
‘𝑍)((𝑄 ×c
𝑂) Func 𝑇)(2nd ‘𝑍)) |
| 178 | 54, 28 | opelxpd 5724 |
. . . . . . . . . 10
⊢ (𝜑 → 〈𝐹, 𝑋〉 ∈ ((𝑂 Func 𝑆) × 𝐵)) |
| 179 | 62, 27 | opelxpd 5724 |
. . . . . . . . . 10
⊢ (𝜑 → 〈𝐺, 𝑃〉 ∈ ((𝑂 Func 𝑆) × 𝐵)) |
| 180 | 165, 166,
167, 177, 178, 179 | funcf2 17913 |
. . . . . . . . 9
⊢ (𝜑 → (〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉):(〈𝐹, 𝑋〉(Hom ‘(𝑄 ×c 𝑂))〈𝐺, 𝑃〉)⟶(((1st
‘𝑍)‘〈𝐹, 𝑋〉)(Hom ‘𝑇)((1st ‘𝑍)‘〈𝐺, 𝑃〉))) |
| 181 | 164, 43, 10, 14, 118, 54, 28, 62, 27, 166 | xpchom2 18231 |
. . . . . . . . . . 11
⊢ (𝜑 → (〈𝐹, 𝑋〉(Hom ‘(𝑄 ×c 𝑂))〈𝐺, 𝑃〉) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃))) |
| 182 | 120 | xpeq2i 5712 |
. . . . . . . . . . 11
⊢ ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋)) |
| 183 | 181, 182 | eqtrdi 2793 |
. . . . . . . . . 10
⊢ (𝜑 → (〈𝐹, 𝑋〉(Hom ‘(𝑄 ×c 𝑂))〈𝐺, 𝑃〉) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))) |
| 184 | | df-ov 7434 |
. . . . . . . . . . . . 13
⊢ (𝐹(1st ‘𝑍)𝑋) = ((1st ‘𝑍)‘〈𝐹, 𝑋〉) |
| 185 | | df-ov 7434 |
. . . . . . . . . . . . 13
⊢ (𝐺(1st ‘𝑍)𝑃) = ((1st ‘𝑍)‘〈𝐺, 𝑃〉) |
| 186 | 184, 185 | oveq12i 7443 |
. . . . . . . . . . . 12
⊢ ((𝐹(1st ‘𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝑍)𝑃)) = (((1st ‘𝑍)‘〈𝐹, 𝑋〉)(Hom ‘𝑇)((1st ‘𝑍)‘〈𝐺, 𝑃〉)) |
| 187 | 186 | eqcomi 2746 |
. . . . . . . . . . 11
⊢
(((1st ‘𝑍)‘〈𝐹, 𝑋〉)(Hom ‘𝑇)((1st ‘𝑍)‘〈𝐺, 𝑃〉)) = ((𝐹(1st ‘𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝑍)𝑃)) |
| 188 | 187 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → (((1st
‘𝑍)‘〈𝐹, 𝑋〉)(Hom ‘𝑇)((1st ‘𝑍)‘〈𝐺, 𝑃〉)) = ((𝐹(1st ‘𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝑍)𝑃))) |
| 189 | 183, 188 | feq23d 6731 |
. . . . . . . . 9
⊢ (𝜑 → ((〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉):(〈𝐹, 𝑋〉(Hom ‘(𝑄 ×c 𝑂))〈𝐺, 𝑃〉)⟶(((1st
‘𝑍)‘〈𝐹, 𝑋〉)(Hom ‘𝑇)((1st ‘𝑍)‘〈𝐺, 𝑃〉)) ↔ (〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉):((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))⟶((𝐹(1st ‘𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝑍)𝑃)))) |
| 190 | 180, 189 | mpbid 232 |
. . . . . . . 8
⊢ (𝜑 → (〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉):((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))⟶((𝐹(1st ‘𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝑍)𝑃))) |
| 191 | 190, 34, 30 | fovcdmd 7605 |
. . . . . . 7
⊢ (𝜑 → (𝐴(〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉)𝐾) ∈ ((𝐹(1st ‘𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝑍)𝑃))) |
| 192 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(Base‘𝑇) =
(Base‘𝑇) |
| 193 | 165, 192,
177 | funcf1 17911 |
. . . . . . . . . 10
⊢ (𝜑 → (1st
‘𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇)) |
| 194 | 193, 54, 28 | fovcdmd 7605 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹(1st ‘𝑍)𝑋) ∈ (Base‘𝑇)) |
| 195 | 169, 19 | setcbas 18123 |
. . . . . . . . 9
⊢ (𝜑 → 𝑉 = (Base‘𝑇)) |
| 196 | 194, 195 | eleqtrrd 2844 |
. . . . . . . 8
⊢ (𝜑 → (𝐹(1st ‘𝑍)𝑋) ∈ 𝑉) |
| 197 | 193, 62, 27 | fovcdmd 7605 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺(1st ‘𝑍)𝑃) ∈ (Base‘𝑇)) |
| 198 | 197, 195 | eleqtrrd 2844 |
. . . . . . . 8
⊢ (𝜑 → (𝐺(1st ‘𝑍)𝑃) ∈ 𝑉) |
| 199 | 169, 19, 167, 196, 198 | elsetchom 18126 |
. . . . . . 7
⊢ (𝜑 → ((𝐴(〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉)𝐾) ∈ ((𝐹(1st ‘𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝑍)𝑃)) ↔ (𝐴(〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉)𝐾):(𝐹(1st ‘𝑍)𝑋)⟶(𝐺(1st ‘𝑍)𝑃))) |
| 200 | 191, 199 | mpbid 232 |
. . . . . 6
⊢ (𝜑 → (𝐴(〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉)𝐾):(𝐹(1st ‘𝑍)𝑋)⟶(𝐺(1st ‘𝑍)𝑃)) |
| 201 | 16, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 54, 28, 62, 27, 34, 30 | yonedalem22 18323 |
. . . . . . . 8
⊢ (𝜑 → (𝐴(〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉)𝐾) = (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)(〈((1st ‘𝑌)‘𝑋), 𝐹〉(2nd ‘𝐻)〈((1st
‘𝑌)‘𝑃), 𝐺〉)𝐴)) |
| 202 | 8 | oppccat 17765 |
. . . . . . . . . . 11
⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
| 203 | 17, 202 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑂 ∈ Cat) |
| 204 | 18 | setccat 18130 |
. . . . . . . . . . 11
⊢ (𝑈 ∈ V → 𝑆 ∈ Cat) |
| 205 | 22, 204 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 ∈ Cat) |
| 206 | 6, 203, 205 | fuccat 18018 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 ∈ Cat) |
| 207 | 170, 206,
43, 14, 45, 54, 82, 62, 12, 31, 34 | hof2val 18301 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)(〈((1st ‘𝑌)‘𝑋), 𝐹〉(2nd ‘𝐻)〈((1st
‘𝑌)‘𝑃), 𝐺〉)𝐴) = (𝑏 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾)))) |
| 208 | 201, 207 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → (𝐴(〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉)𝐾) = (𝑏 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾)))) |
| 209 | 16, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 54, 28 | yonedalem21 18318 |
. . . . . . 7
⊢ (𝜑 → (𝐹(1st ‘𝑍)𝑋) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) |
| 210 | 16, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 62, 27 | yonedalem21 18318 |
. . . . . . 7
⊢ (𝜑 → (𝐺(1st ‘𝑍)𝑃) = (((1st ‘𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺)) |
| 211 | 208, 209,
210 | feq123d 6725 |
. . . . . 6
⊢ (𝜑 → ((𝐴(〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉)𝐾):(𝐹(1st ‘𝑍)𝑋)⟶(𝐺(1st ‘𝑍)𝑃) ↔ (𝑏 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))):(((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶(((1st ‘𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺))) |
| 212 | 200, 211 | mpbid 232 |
. . . . 5
⊢ (𝜑 → (𝑏 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))):(((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶(((1st ‘𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺)) |
| 213 | | eqid 2737 |
. . . . . 6
⊢ (𝑏 ∈ (((1st
‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))) = (𝑏 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))) |
| 214 | 213 | fmpt 7130 |
. . . . 5
⊢
(∀𝑏 ∈
(((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾)) ∈ (((1st ‘𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↔ (𝑏 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))):(((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶(((1st ‘𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺)) |
| 215 | 212, 214 | sylibr 234 |
. . . 4
⊢ (𝜑 → ∀𝑏 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾)) ∈ (((1st ‘𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺)) |
| 216 | | yonedalem3.m |
. . . . . 6
⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥)))) |
| 217 | 16, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 62, 27, 216 | yonedalem3a 18319 |
. . . . 5
⊢ (𝜑 → ((𝐺𝑀𝑃) = (𝑎 ∈ (((1st ‘𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↦ ((𝑎‘𝑃)‘( 1 ‘𝑃))) ∧ (𝐺𝑀𝑃):(𝐺(1st ‘𝑍)𝑃)⟶(𝐺(1st ‘𝐸)𝑃))) |
| 218 | 217 | simpld 494 |
. . . 4
⊢ (𝜑 → (𝐺𝑀𝑃) = (𝑎 ∈ (((1st ‘𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↦ ((𝑎‘𝑃)‘( 1 ‘𝑃)))) |
| 219 | | fveq1 6905 |
. . . . 5
⊢ (𝑎 = ((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾)) → (𝑎‘𝑃) = (((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃)) |
| 220 | 219 | fveq1d 6908 |
. . . 4
⊢ (𝑎 = ((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾)) → ((𝑎‘𝑃)‘( 1 ‘𝑃)) = ((((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃)‘( 1 ‘𝑃))) |
| 221 | 215, 208,
218, 220 | fmptcof 7150 |
. . 3
⊢ (𝜑 → ((𝐺𝑀𝑃) ∘ (𝐴(〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉)𝐾)) = (𝑏 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(〈((1st ‘𝑌)‘𝑋), 𝐹〉(comp‘𝑄)𝐺)𝑏)(〈((1st ‘𝑌)‘𝑃), ((1st ‘𝑌)‘𝑋)〉(comp‘𝑄)𝐺)((𝑃(2nd ‘𝑌)𝑋)‘𝐾))‘𝑃)‘( 1 ‘𝑃)))) |
| 222 | | eqid 2737 |
. . . . . . 7
⊢
(〈𝐹, 𝑋〉(2nd
‘𝐸)〈𝐺, 𝑃〉) = (〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉) |
| 223 | 172, 203,
205, 10, 118, 11, 7, 54, 62, 28, 27, 222, 34, 121 | evlf2val 18264 |
. . . . . 6
⊢ (𝜑 → (𝐴(〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉)𝐾) = ((𝐴‘𝑃)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑃)〉(comp‘𝑆)((1st ‘𝐺)‘𝑃))((𝑋(2nd ‘𝐹)𝑃)‘𝐾))) |
| 224 | 18, 22, 11, 137, 60, 67, 145, 77 | setcco 18128 |
. . . . . 6
⊢ (𝜑 → ((𝐴‘𝑃)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑃)〉(comp‘𝑆)((1st ‘𝐺)‘𝑃))((𝑋(2nd ‘𝐹)𝑃)‘𝐾)) = ((𝐴‘𝑃) ∘ ((𝑋(2nd ‘𝐹)𝑃)‘𝐾))) |
| 225 | 223, 224 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → (𝐴(〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉)𝐾) = ((𝐴‘𝑃) ∘ ((𝑋(2nd ‘𝐹)𝑃)‘𝐾))) |
| 226 | 225 | coeq1d 5872 |
. . . 4
⊢ (𝜑 → ((𝐴(〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉)𝐾) ∘ (𝐹𝑀𝑋)) = (((𝐴‘𝑃) ∘ ((𝑋(2nd ‘𝐹)𝑃)‘𝐾)) ∘ (𝐹𝑀𝑋))) |
| 227 | 16, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 54, 28, 216 | yonedalem3a 18319 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹𝑀𝑋) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))) ∧ (𝐹𝑀𝑋):(𝐹(1st ‘𝑍)𝑋)⟶(𝐹(1st ‘𝐸)𝑋))) |
| 228 | 227 | simprd 495 |
. . . . . . 7
⊢ (𝜑 → (𝐹𝑀𝑋):(𝐹(1st ‘𝑍)𝑋)⟶(𝐹(1st ‘𝐸)𝑋)) |
| 229 | 227 | simpld 494 |
. . . . . . . 8
⊢ (𝜑 → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋)))) |
| 230 | 172, 203,
205, 10, 54, 28 | evlf1 18265 |
. . . . . . . 8
⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋)) |
| 231 | 229, 209,
230 | feq123d 6725 |
. . . . . . 7
⊢ (𝜑 → ((𝐹𝑀𝑋):(𝐹(1st ‘𝑍)𝑋)⟶(𝐹(1st ‘𝐸)𝑋) ↔ (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))):(((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st ‘𝐹)‘𝑋))) |
| 232 | 228, 231 | mpbid 232 |
. . . . . 6
⊢ (𝜑 → (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))):(((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st ‘𝐹)‘𝑋)) |
| 233 | | eqid 2737 |
. . . . . . 7
⊢ (𝑎 ∈ (((1st
‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))) |
| 234 | 233 | fmpt 7130 |
. . . . . 6
⊢
(∀𝑎 ∈
(((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)((𝑎‘𝑋)‘( 1 ‘𝑋)) ∈ ((1st ‘𝐹)‘𝑋) ↔ (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))):(((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st ‘𝐹)‘𝑋)) |
| 235 | 232, 234 | sylibr 234 |
. . . . 5
⊢ (𝜑 → ∀𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)((𝑎‘𝑋)‘( 1 ‘𝑋)) ∈ ((1st ‘𝐹)‘𝑋)) |
| 236 | | fcompt 7153 |
. . . . . 6
⊢ (((𝐴‘𝑃):((1st ‘𝐹)‘𝑃)⟶((1st ‘𝐺)‘𝑃) ∧ ((𝑋(2nd ‘𝐹)𝑃)‘𝐾):((1st ‘𝐹)‘𝑋)⟶((1st ‘𝐹)‘𝑃)) → ((𝐴‘𝑃) ∘ ((𝑋(2nd ‘𝐹)𝑃)‘𝐾)) = (𝑦 ∈ ((1st ‘𝐹)‘𝑋) ↦ ((𝐴‘𝑃)‘(((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘𝑦)))) |
| 237 | 77, 145, 236 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ((𝐴‘𝑃) ∘ ((𝑋(2nd ‘𝐹)𝑃)‘𝐾)) = (𝑦 ∈ ((1st ‘𝐹)‘𝑋) ↦ ((𝐴‘𝑃)‘(((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘𝑦)))) |
| 238 | | 2fveq3 6911 |
. . . . 5
⊢ (𝑦 = ((𝑎‘𝑋)‘( 1 ‘𝑋)) → ((𝐴‘𝑃)‘(((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘𝑦)) = ((𝐴‘𝑃)‘(((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘((𝑎‘𝑋)‘( 1 ‘𝑋))))) |
| 239 | 235, 229,
237, 238 | fmptcof 7150 |
. . . 4
⊢ (𝜑 → (((𝐴‘𝑃) ∘ ((𝑋(2nd ‘𝐹)𝑃)‘𝐾)) ∘ (𝐹𝑀𝑋)) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴‘𝑃)‘(((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘((𝑎‘𝑋)‘( 1 ‘𝑋)))))) |
| 240 | 226, 239 | eqtrd 2777 |
. . 3
⊢ (𝜑 → ((𝐴(〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉)𝐾) ∘ (𝐹𝑀𝑋)) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴‘𝑃)‘(((𝑋(2nd ‘𝐹)𝑃)‘𝐾)‘((𝑎‘𝑋)‘( 1 ‘𝑋)))))) |
| 241 | 163, 221,
240 | 3eqtr4d 2787 |
. 2
⊢ (𝜑 → ((𝐺𝑀𝑃) ∘ (𝐴(〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉)𝐾)) = ((𝐴(〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉)𝐾) ∘ (𝐹𝑀𝑋))) |
| 242 | | eqid 2737 |
. . 3
⊢
(comp‘𝑇) =
(comp‘𝑇) |
| 243 | 174 | simprd 495 |
. . . . . . 7
⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
| 244 | | 1st2ndbr 8067 |
. . . . . . 7
⊢ ((Rel
((𝑄
×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st ‘𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd ‘𝐸)) |
| 245 | 168, 243,
244 | sylancr 587 |
. . . . . 6
⊢ (𝜑 → (1st
‘𝐸)((𝑄 ×c
𝑂) Func 𝑇)(2nd ‘𝐸)) |
| 246 | 165, 192,
245 | funcf1 17911 |
. . . . 5
⊢ (𝜑 → (1st
‘𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇)) |
| 247 | 246, 62, 27 | fovcdmd 7605 |
. . . 4
⊢ (𝜑 → (𝐺(1st ‘𝐸)𝑃) ∈ (Base‘𝑇)) |
| 248 | 247, 195 | eleqtrrd 2844 |
. . 3
⊢ (𝜑 → (𝐺(1st ‘𝐸)𝑃) ∈ 𝑉) |
| 249 | 217 | simprd 495 |
. . 3
⊢ (𝜑 → (𝐺𝑀𝑃):(𝐺(1st ‘𝑍)𝑃)⟶(𝐺(1st ‘𝐸)𝑃)) |
| 250 | 169, 19, 242, 196, 198, 248, 200, 249 | setcco 18128 |
. 2
⊢ (𝜑 → ((𝐺𝑀𝑃)(〈(𝐹(1st ‘𝑍)𝑋), (𝐺(1st ‘𝑍)𝑃)〉(comp‘𝑇)(𝐺(1st ‘𝐸)𝑃))(𝐴(〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉)𝐾)) = ((𝐺𝑀𝑃) ∘ (𝐴(〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉)𝐾))) |
| 251 | 246, 54, 28 | fovcdmd 7605 |
. . . 4
⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) ∈ (Base‘𝑇)) |
| 252 | 251, 195 | eleqtrrd 2844 |
. . 3
⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) ∈ 𝑉) |
| 253 | 165, 166,
167, 245, 178, 179 | funcf2 17913 |
. . . . . 6
⊢ (𝜑 → (〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉):(〈𝐹, 𝑋〉(Hom ‘(𝑄 ×c 𝑂))〈𝐺, 𝑃〉)⟶(((1st
‘𝐸)‘〈𝐹, 𝑋〉)(Hom ‘𝑇)((1st ‘𝐸)‘〈𝐺, 𝑃〉))) |
| 254 | | df-ov 7434 |
. . . . . . . . . 10
⊢ (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐸)‘〈𝐹, 𝑋〉) |
| 255 | | df-ov 7434 |
. . . . . . . . . 10
⊢ (𝐺(1st ‘𝐸)𝑃) = ((1st ‘𝐸)‘〈𝐺, 𝑃〉) |
| 256 | 254, 255 | oveq12i 7443 |
. . . . . . . . 9
⊢ ((𝐹(1st ‘𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝐸)𝑃)) = (((1st ‘𝐸)‘〈𝐹, 𝑋〉)(Hom ‘𝑇)((1st ‘𝐸)‘〈𝐺, 𝑃〉)) |
| 257 | 256 | eqcomi 2746 |
. . . . . . . 8
⊢
(((1st ‘𝐸)‘〈𝐹, 𝑋〉)(Hom ‘𝑇)((1st ‘𝐸)‘〈𝐺, 𝑃〉)) = ((𝐹(1st ‘𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝐸)𝑃)) |
| 258 | 257 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (((1st
‘𝐸)‘〈𝐹, 𝑋〉)(Hom ‘𝑇)((1st ‘𝐸)‘〈𝐺, 𝑃〉)) = ((𝐹(1st ‘𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝐸)𝑃))) |
| 259 | 183, 258 | feq23d 6731 |
. . . . . 6
⊢ (𝜑 → ((〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉):(〈𝐹, 𝑋〉(Hom ‘(𝑄 ×c 𝑂))〈𝐺, 𝑃〉)⟶(((1st
‘𝐸)‘〈𝐹, 𝑋〉)(Hom ‘𝑇)((1st ‘𝐸)‘〈𝐺, 𝑃〉)) ↔ (〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉):((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))⟶((𝐹(1st ‘𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝐸)𝑃)))) |
| 260 | 253, 259 | mpbid 232 |
. . . . 5
⊢ (𝜑 → (〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉):((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))⟶((𝐹(1st ‘𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝐸)𝑃))) |
| 261 | 260, 34, 30 | fovcdmd 7605 |
. . . 4
⊢ (𝜑 → (𝐴(〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉)𝐾) ∈ ((𝐹(1st ‘𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝐸)𝑃))) |
| 262 | 169, 19, 167, 252, 248 | elsetchom 18126 |
. . . 4
⊢ (𝜑 → ((𝐴(〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉)𝐾) ∈ ((𝐹(1st ‘𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st ‘𝐸)𝑃)) ↔ (𝐴(〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉)𝐾):(𝐹(1st ‘𝐸)𝑋)⟶(𝐺(1st ‘𝐸)𝑃))) |
| 263 | 261, 262 | mpbid 232 |
. . 3
⊢ (𝜑 → (𝐴(〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉)𝐾):(𝐹(1st ‘𝐸)𝑋)⟶(𝐺(1st ‘𝐸)𝑃)) |
| 264 | 169, 19, 242, 196, 252, 248, 228, 263 | setcco 18128 |
. 2
⊢ (𝜑 → ((𝐴(〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉)𝐾)(〈(𝐹(1st ‘𝑍)𝑋), (𝐹(1st ‘𝐸)𝑋)〉(comp‘𝑇)(𝐺(1st ‘𝐸)𝑃))(𝐹𝑀𝑋)) = ((𝐴(〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉)𝐾) ∘ (𝐹𝑀𝑋))) |
| 265 | 241, 250,
264 | 3eqtr4d 2787 |
1
⊢ (𝜑 → ((𝐺𝑀𝑃)(〈(𝐹(1st ‘𝑍)𝑋), (𝐺(1st ‘𝑍)𝑃)〉(comp‘𝑇)(𝐺(1st ‘𝐸)𝑃))(𝐴(〈𝐹, 𝑋〉(2nd ‘𝑍)〈𝐺, 𝑃〉)𝐾)) = ((𝐴(〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑃〉)𝐾)(〈(𝐹(1st ‘𝑍)𝑋), (𝐹(1st ‘𝐸)𝑋)〉(comp‘𝑇)(𝐺(1st ‘𝐸)𝑃))(𝐹𝑀𝑋))) |