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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 18179 | . 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 4880 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 〈𝑋, 𝑧〉 = 〈𝑋, 𝑍〉) |
16 | 14 | opeq2d 4880 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 〈𝑌, 𝑧〉 = 〈𝑌, 𝑍〉) |
17 | 15, 16 | oveq12d 7424 | . . 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 6893 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝐼‘𝑧) = (𝐼‘𝑍)) |
20 | 17, 18, 19 | oveq123d 7427 | . 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 7441 | . 2 ⊢ (𝜑 → (𝐾(〈𝑋, 𝑍〉(2nd ‘𝐹)〈𝑌, 𝑍〉)(𝐼‘𝑍)) ∈ V) | |
23 | 13, 20, 21, 22 | fvmptd 7003 | 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 4634 ‘cfv 6541 (class class class)co 7406 2nd c2nd 7971 Basecbs 17141 Hom chom 17205 Catccat 17605 Idccid 17606 Func cfunc 17801 ×c cxpc 18117 curryF ccurf 18160 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-1st 7972 df-2nd 7973 df-curf 18164 |
This theorem is referenced by: curf2cl 18181 curfcl 18182 uncfcurf 18189 yon2 18216 |
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