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Mirrors > Home > MPE Home > Th. List > curf2val | Structured version Visualization version GIF version |
Description: Value of a component of the curry functor natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.) |
Ref | Expression |
---|---|
curf2.g | ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹) |
curf2.a | ⊢ 𝐴 = (Base‘𝐶) |
curf2.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
curf2.d | ⊢ (𝜑 → 𝐷 ∈ Cat) |
curf2.f | ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
curf2.b | ⊢ 𝐵 = (Base‘𝐷) |
curf2.h | ⊢ 𝐻 = (Hom ‘𝐶) |
curf2.i | ⊢ 𝐼 = (Id‘𝐷) |
curf2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
curf2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐴) |
curf2.k | ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) |
curf2.l | ⊢ 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾) |
curf2.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
curf2val | ⊢ (𝜑 → (𝐿‘𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | curf2.g | . . 3 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹) | |
2 | curf2.a | . . 3 ⊢ 𝐴 = (Base‘𝐶) | |
3 | curf2.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | curf2.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
5 | curf2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) | |
6 | curf2.b | . . 3 ⊢ 𝐵 = (Base‘𝐷) | |
7 | curf2.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
8 | curf2.i | . . 3 ⊢ 𝐼 = (Id‘𝐷) | |
9 | curf2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
10 | curf2.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴) | |
11 | curf2.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) | |
12 | curf2.l | . . 3 ⊢ 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | curf2 18182 | . 2 ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)))) |
14 | simpr 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍) | |
15 | 14 | opeq2d 4881 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩) |
16 | 14 | opeq2d 4881 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ⟨𝑌, 𝑧⟩ = ⟨𝑌, 𝑍⟩) |
17 | 15, 16 | oveq12d 7427 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩) = (⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)) |
18 | eqidd 2734 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 = 𝐾) | |
19 | 14 | fveq2d 6896 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝐼‘𝑧) = (𝐼‘𝑍)) |
20 | 17, 18, 19 | oveq123d 7430 | . 2 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)) = (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍))) |
21 | curf2.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
22 | ovexd 7444 | . 2 ⊢ (𝜑 → (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍)) ∈ V) | |
23 | 13, 20, 21, 22 | fvmptd 7006 | 1 ⊢ (𝜑 → (𝐿‘𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⟨cop 4635 ‘cfv 6544 (class class class)co 7409 2nd c2nd 7974 Basecbs 17144 Hom chom 17208 Catccat 17608 Idccid 17609 Func cfunc 17804 ×c cxpc 18120 curryF ccurf 18163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-curf 18167 |
This theorem is referenced by: curf2cl 18184 curfcl 18185 uncfcurf 18192 yon2 18219 |
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