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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 2731 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
10 | eqid 2731 | . . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | curf1 18183 | . . 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 6905 | . . . . 5 ⊢ 𝐵 ∈ V |
13 | 12 | mptex 7227 | . . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)) ∈ V |
14 | 12, 12 | mpoex 8070 | . . . 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 7989 | . . 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 7428 | . 2 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝑋(1st ‘𝐹)𝑦) = (𝑋(1st ‘𝐹)𝑌)) |
19 | curf11.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
20 | ovexd 7447 | . 2 ⊢ (𝜑 → (𝑋(1st ‘𝐹)𝑌) ∈ V) | |
21 | 16, 18, 19, 20 | fvmptd 7005 | 1 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = (𝑋(1st ‘𝐹)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 〈cop 4634 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 1st c1st 7977 2nd c2nd 7978 Basecbs 17149 Hom chom 17213 Catccat 17613 Idccid 17614 Func cfunc 17809 ×c cxpc 18125 curryF ccurf 18168 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-curf 18172 |
This theorem is referenced by: curf1cl 18186 curf2cl 18189 curfcl 18190 uncfcurf 18197 diag11 18201 yon11 18222 |
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