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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 2610 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
5 | 1, 2, 3, 4 | idfuval 16359 | . . 3 ⊢ (𝜑 → 𝐼 = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))〉) |
6 | 5 | fveq2d 6107 | . 2 ⊢ (𝜑 → (1st ‘𝐼) = (1st ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))〉)) |
7 | fvex 6113 | . . . . 5 ⊢ (Base‘𝐶) ∈ V | |
8 | 2, 7 | eqeltri 2684 | . . . 4 ⊢ 𝐵 ∈ V |
9 | resiexg 6994 | . . . 4 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ( I ↾ 𝐵) ∈ V |
11 | 8, 8 | xpex 6860 | . . . 4 ⊢ (𝐵 × 𝐵) ∈ V |
12 | 11 | mptex 6390 | . . 3 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V |
13 | 10, 12 | op1st 7067 | . 2 ⊢ (1st ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))〉) = ( I ↾ 𝐵) |
14 | 6, 13 | syl6eq 2660 | 1 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ↦ cmpt 4643 I cid 4948 × cxp 5036 ↾ cres 5040 ‘cfv 5804 1st c1st 7057 Basecbs 15695 Hom chom 15779 Catccat 16148 idfunccidfu 16338 |
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-1st 7059 df-idfu 16342 |
This theorem is referenced by: idfu1 16363 cofulid 16373 cofurid 16374 catciso 16580 curf2ndf 16710 |
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