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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 18230 | . 2 ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)))) |
14 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍) | |
15 | 14 | opeq2d 4885 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩) |
16 | 14 | opeq2d 4885 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ⟨𝑌, 𝑧⟩ = ⟨𝑌, 𝑍⟩) |
17 | 15, 16 | oveq12d 7444 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩) = (⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)) |
18 | eqidd 2729 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 = 𝐾) | |
19 | 14 | fveq2d 6906 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝐼‘𝑧) = (𝐼‘𝑍)) |
20 | 17, 18, 19 | oveq123d 7447 | . 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 7461 | . 2 ⊢ (𝜑 → (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍)) ∈ V) | |
23 | 13, 20, 21, 22 | fvmptd 7017 | 1 ⊢ (𝜑 → (𝐿‘𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⟨cop 4638 ‘cfv 6553 (class class class)co 7426 2nd c2nd 8000 Basecbs 17189 Hom chom 17253 Catccat 17653 Idccid 17654 Func cfunc 17849 ×c cxpc 18168 curryF ccurf 18211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-curf 18215 |
This theorem is referenced by: curf2cl 18232 curfcl 18233 uncfcurf 18240 yon2 18267 |
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