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Mirrors > Home > MPE Home > Th. List > curf11 | Structured version Visualization version GIF version |
Description: Value of the double evaluated curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
curfval.g | ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹) |
curfval.a | ⊢ 𝐴 = (Base‘𝐶) |
curfval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
curfval.d | ⊢ (𝜑 → 𝐷 ∈ Cat) |
curfval.f | ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
curfval.b | ⊢ 𝐵 = (Base‘𝐷) |
curf1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
curf1.k | ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋) |
curf11.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
curf11 | ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = (𝑋(1st ‘𝐹)𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | curfval.g | . . . 4 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹) | |
2 | curfval.a | . . . 4 ⊢ 𝐴 = (Base‘𝐶) | |
3 | curfval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | curfval.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
5 | curfval.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) | |
6 | curfval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐷) | |
7 | curf1.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
8 | curf1.k | . . . 4 ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋) | |
9 | eqid 2728 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
10 | eqid 2728 | . . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | curf1 18217 | . . 3 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩) |
12 | 6 | fvexi 6911 | . . . . 5 ⊢ 𝐵 ∈ V |
13 | 12 | mptex 7235 | . . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)) ∈ V |
14 | 12, 12 | mpoex 8084 | . . . 4 ⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V |
15 | 13, 14 | op1std 8003 | . . 3 ⊢ (𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦))) |
16 | 11, 15 | syl 17 | . 2 ⊢ (𝜑 → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦))) |
17 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
18 | 17 | oveq2d 7436 | . 2 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝑋(1st ‘𝐹)𝑦) = (𝑋(1st ‘𝐹)𝑌)) |
19 | curf11.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
20 | ovexd 7455 | . 2 ⊢ (𝜑 → (𝑋(1st ‘𝐹)𝑌) ∈ V) | |
21 | 16, 18, 19, 20 | fvmptd 7012 | 1 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = (𝑋(1st ‘𝐹)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⟨cop 4635 ↦ cmpt 5231 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 1st c1st 7991 2nd c2nd 7992 Basecbs 17180 Hom chom 17244 Catccat 17644 Idccid 17645 Func cfunc 17840 ×c cxpc 18159 curryF ccurf 18202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-curf 18206 |
This theorem is referenced by: curf1cl 18220 curf2cl 18223 curfcl 18224 uncfcurf 18231 diag11 18235 yon11 18256 |
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