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Mirrors > Home > MPE Home > Th. List > idfu1st | Structured version Visualization version GIF version |
Description: Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
idfuval.i | ⊢ 𝐼 = (idfunc‘𝐶) |
idfuval.b | ⊢ 𝐵 = (Base‘𝐶) |
idfuval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
Ref | Expression |
---|---|
idfu1st | ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idfuval.i | . . . 4 ⊢ 𝐼 = (idfunc‘𝐶) | |
2 | idfuval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | idfuval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | eqid 2735 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
5 | 1, 2, 3, 4 | idfuval 17927 | . . 3 ⊢ (𝜑 → 𝐼 = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))〉) |
6 | 5 | fveq2d 6911 | . 2 ⊢ (𝜑 → (1st ‘𝐼) = (1st ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))〉)) |
7 | 2 | fvexi 6921 | . . . 4 ⊢ 𝐵 ∈ V |
8 | resiexg 7935 | . . . 4 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ ( I ↾ 𝐵) ∈ V |
10 | 7, 7 | xpex 7772 | . . . 4 ⊢ (𝐵 × 𝐵) ∈ V |
11 | 10 | mptex 7243 | . . 3 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V |
12 | 9, 11 | op1st 8021 | . 2 ⊢ (1st ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))〉) = ( I ↾ 𝐵) |
13 | 6, 12 | eqtrdi 2791 | 1 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ↦ cmpt 5231 I cid 5582 × cxp 5687 ↾ cres 5691 ‘cfv 6563 1st c1st 8011 Basecbs 17245 Hom chom 17309 Catccat 17709 idfunccidfu 17906 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-1st 8013 df-idfu 17910 |
This theorem is referenced by: idfu1 17931 cofulid 17941 cofurid 17942 catciso 18165 curf2ndf 18304 |
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