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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 2821 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
5 | 1, 2, 3, 4 | idfuval 17146 | . . 3 ⊢ (𝜑 → 𝐼 = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))〉) |
6 | 5 | fveq2d 6674 | . 2 ⊢ (𝜑 → (1st ‘𝐼) = (1st ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))〉)) |
7 | 2 | fvexi 6684 | . . . 4 ⊢ 𝐵 ∈ V |
8 | resiexg 7619 | . . . 4 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ ( I ↾ 𝐵) ∈ V |
10 | 7, 7 | xpex 7476 | . . . 4 ⊢ (𝐵 × 𝐵) ∈ V |
11 | 10 | mptex 6986 | . . 3 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V |
12 | 9, 11 | op1st 7697 | . 2 ⊢ (1st ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))〉) = ( I ↾ 𝐵) |
13 | 6, 12 | syl6eq 2872 | 1 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 ↦ cmpt 5146 I cid 5459 × cxp 5553 ↾ cres 5557 ‘cfv 6355 1st c1st 7687 Basecbs 16483 Hom chom 16576 Catccat 16935 idfunccidfu 17125 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-1st 7689 df-idfu 17129 |
This theorem is referenced by: idfu1 17150 cofulid 17160 cofurid 17161 catciso 17367 curf2ndf 17497 |
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