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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 18274 | . 2 ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧)))) |
| 14 | simpr 484 | . . . . 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 7449 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉) = (〈𝑋, 𝑍〉(2nd ‘𝐹)〈𝑌, 𝑍〉)) |
| 18 | eqidd 2738 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 = 𝐾) | |
| 19 | 14 | fveq2d 6910 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝐼‘𝑧) = (𝐼‘𝑍)) |
| 20 | 17, 18, 19 | oveq123d 7452 | . 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 7466 | . 2 ⊢ (𝜑 → (𝐾(〈𝑋, 𝑍〉(2nd ‘𝐹)〈𝑌, 𝑍〉)(𝐼‘𝑍)) ∈ V) | |
| 23 | 13, 20, 21, 22 | fvmptd 7023 | 1 ⊢ (𝜑 → (𝐿‘𝑍) = (𝐾(〈𝑋, 𝑍〉(2nd ‘𝐹)〈𝑌, 𝑍〉)(𝐼‘𝑍))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ‘cfv 6561 (class class class)co 7431 2nd c2nd 8013 Basecbs 17247 Hom chom 17308 Catccat 17707 Idccid 17708 Func cfunc 17899 ×c cxpc 18213 curryF ccurf 18255 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-curf 18259 |
| This theorem is referenced by: curf2cl 18276 curfcl 18277 uncfcurf 18284 yon2 18311 |
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