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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 2821 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
10 | eqid 2821 | . . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | curf1 17475 | . . 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 6684 | . . . . 5 ⊢ 𝐵 ∈ V |
13 | 12 | mptex 6986 | . . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)) ∈ V |
14 | 12, 12 | mpoex 7777 | . . . 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 7699 | . . 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 487 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
18 | 17 | oveq2d 7172 | . 2 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝑋(1st ‘𝐹)𝑦) = (𝑋(1st ‘𝐹)𝑌)) |
19 | curf11.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
20 | ovexd 7191 | . 2 ⊢ (𝜑 → (𝑋(1st ‘𝐹)𝑌) ∈ V) | |
21 | 16, 18, 19, 20 | fvmptd 6775 | 1 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = (𝑋(1st ‘𝐹)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 1st c1st 7687 2nd c2nd 7688 Basecbs 16483 Hom chom 16576 Catccat 16935 Idccid 16936 Func cfunc 17124 ×c cxpc 17418 curryF ccurf 17460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-curf 17464 |
This theorem is referenced by: curf1cl 17478 curf2cl 17481 curfcl 17482 uncfcurf 17489 diag11 17493 yon11 17514 |
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