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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 16692 | . 2 ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧)))) |
14 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍) | |
15 | 14 | opeq2d 4347 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 〈𝑋, 𝑧〉 = 〈𝑋, 𝑍〉) |
16 | 14 | opeq2d 4347 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 〈𝑌, 𝑧〉 = 〈𝑌, 𝑍〉) |
17 | 15, 16 | oveq12d 6567 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉) = (〈𝑋, 𝑍〉(2nd ‘𝐹)〈𝑌, 𝑍〉)) |
18 | eqidd 2611 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 = 𝐾) | |
19 | 14 | fveq2d 6107 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝐼‘𝑧) = (𝐼‘𝑍)) |
20 | 17, 18, 19 | oveq123d 6570 | . 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 | ovex 6577 | . . 3 ⊢ (𝐾(〈𝑋, 𝑍〉(2nd ‘𝐹)〈𝑌, 𝑍〉)(𝐼‘𝑍)) ∈ V | |
23 | 22 | a1i 11 | . 2 ⊢ (𝜑 → (𝐾(〈𝑋, 𝑍〉(2nd ‘𝐹)〈𝑌, 𝑍〉)(𝐼‘𝑍)) ∈ V) |
24 | 13, 20, 21, 23 | fvmptd 6197 | 1 ⊢ (𝜑 → (𝐿‘𝑍) = (𝐾(〈𝑋, 𝑍〉(2nd ‘𝐹)〈𝑌, 𝑍〉)(𝐼‘𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ‘cfv 5804 (class class class)co 6549 2nd c2nd 7058 Basecbs 15695 Hom chom 15779 Catccat 16148 Idccid 16149 Func cfunc 16337 ×c cxpc 16631 curryF ccurf 16673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-curf 16677 |
This theorem is referenced by: curf2cl 16694 curfcl 16695 uncfcurf 16702 yon2 16729 |
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