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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 17469 | . . 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 6678 | . . . . 5 ⊢ 𝐵 ∈ V |
13 | 12 | mptex 6980 | . . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)) ∈ V |
14 | 12, 12 | mpoex 7771 | . . . 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 7693 | . . 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 7166 | . 2 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝑋(1st ‘𝐹)𝑦) = (𝑋(1st ‘𝐹)𝑌)) |
19 | curf11.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
20 | ovexd 7185 | . 2 ⊢ (𝜑 → (𝑋(1st ‘𝐹)𝑌) ∈ V) | |
21 | 16, 18, 19, 20 | fvmptd 6769 | 1 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = (𝑋(1st ‘𝐹)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cop 4566 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 1st c1st 7681 2nd c2nd 7682 Basecbs 16477 Hom chom 16570 Catccat 16929 Idccid 16930 Func cfunc 17118 ×c cxpc 17412 curryF ccurf 17454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-curf 17458 |
This theorem is referenced by: curf1cl 17472 curf2cl 17475 curfcl 17476 uncfcurf 17483 diag11 17487 yon11 17508 |
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