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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 2726 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
10 | eqid 2726 | . . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | curf1 18188 | . . 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 6898 | . . . . 5 ⊢ 𝐵 ∈ V |
13 | 12 | mptex 7219 | . . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)) ∈ V |
14 | 12, 12 | mpoex 8062 | . . . 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 7981 | . . 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 7420 | . 2 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝑋(1st ‘𝐹)𝑦) = (𝑋(1st ‘𝐹)𝑌)) |
19 | curf11.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
20 | ovexd 7439 | . 2 ⊢ (𝜑 → (𝑋(1st ‘𝐹)𝑌) ∈ V) | |
21 | 16, 18, 19, 20 | fvmptd 6998 | 1 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = (𝑋(1st ‘𝐹)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⟨cop 4629 ↦ cmpt 5224 ‘cfv 6536 (class class class)co 7404 ∈ cmpo 7406 1st c1st 7969 2nd c2nd 7970 Basecbs 17151 Hom chom 17215 Catccat 17615 Idccid 17616 Func cfunc 17811 ×c cxpc 18130 curryF ccurf 18173 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-curf 18177 |
This theorem is referenced by: curf1cl 18191 curf2cl 18194 curfcl 18195 uncfcurf 18202 diag11 18206 yon11 18227 |
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