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Mirrors > Home > MPE Home > Th. List > curf1fval | Structured version Visualization version GIF version |
Description: Value of the object part of the 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) |
curfval.j | ⊢ 𝐽 = (Hom ‘𝐷) |
curfval.1 | ⊢ 1 = (Id‘𝐶) |
Ref | Expression |
---|---|
curf1fval | ⊢ (𝜑 → (1st ‘𝐺) = (𝑥 ∈ 𝐴 ↦ 〈(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑧〉)𝑔)))〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | curfval.g | . . 3 ⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) | |
2 | curfval.a | . . 3 ⊢ 𝐴 = (Base‘𝐶) | |
3 | curfval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | curfval.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
5 | curfval.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) | |
6 | curfval.b | . . 3 ⊢ 𝐵 = (Base‘𝐷) | |
7 | curfval.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐷) | |
8 | curfval.1 | . . 3 ⊢ 1 = (Id‘𝐶) | |
9 | eqid 2610 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
10 | eqid 2610 | . . 3 ⊢ (Id‘𝐷) = (Id‘𝐷) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | curfval 16686 | . 2 ⊢ (𝜑 → 𝐺 = 〈(𝑥 ∈ 𝐴 ↦ 〈(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑧〉)𝑔)))〉), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(〈𝑥, 𝑧〉(2nd ‘𝐹)〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧)))))〉) |
12 | fvex 6113 | . . . . 5 ⊢ (Base‘𝐶) ∈ V | |
13 | 2, 12 | eqeltri 2684 | . . . 4 ⊢ 𝐴 ∈ V |
14 | 13 | mptex 6390 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 〈(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑧〉)𝑔)))〉) ∈ V |
15 | 13, 13 | mpt2ex 7136 | . . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(〈𝑥, 𝑧〉(2nd ‘𝐹)〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧))))) ∈ V |
16 | 14, 15 | op1std 7069 | . 2 ⊢ (𝐺 = 〈(𝑥 ∈ 𝐴 ↦ 〈(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑧〉)𝑔)))〉), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(〈𝑥, 𝑧〉(2nd ‘𝐹)〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧)))))〉 → (1st ‘𝐺) = (𝑥 ∈ 𝐴 ↦ 〈(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑧〉)𝑔)))〉)) |
17 | 11, 16 | syl 17 | 1 ⊢ (𝜑 → (1st ‘𝐺) = (𝑥 ∈ 𝐴 ↦ 〈(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑧〉)𝑔)))〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1st c1st 7057 2nd c2nd 7058 Basecbs 15695 Hom chom 15779 Catccat 16148 Idccid 16149 Func cfunc 16337 ×c cxpc 16631 curryF ccurf 16673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-curf 16677 |
This theorem is referenced by: curf1 16688 |
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