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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 2823 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
5 | 1, 2, 3, 4 | idfuval 17148 | . . 3 ⊢ (𝜑 → 𝐼 = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))〉) |
6 | 5 | fveq2d 6676 | . 2 ⊢ (𝜑 → (1st ‘𝐼) = (1st ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))〉)) |
7 | 2 | fvexi 6686 | . . . 4 ⊢ 𝐵 ∈ V |
8 | resiexg 7621 | . . . 4 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ ( I ↾ 𝐵) ∈ V |
10 | 7, 7 | xpex 7478 | . . . 4 ⊢ (𝐵 × 𝐵) ∈ V |
11 | 10 | mptex 6988 | . . 3 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧))) ∈ V |
12 | 9, 11 | op1st 7699 | . 2 ⊢ (1st ‘〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐶)‘𝑧)))〉) = ( I ↾ 𝐵) |
13 | 6, 12 | syl6eq 2874 | 1 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 〈cop 4575 ↦ cmpt 5148 I cid 5461 × cxp 5555 ↾ cres 5559 ‘cfv 6357 1st c1st 7689 Basecbs 16485 Hom chom 16578 Catccat 16937 idfunccidfu 17127 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-1st 7691 df-idfu 17131 |
This theorem is referenced by: idfu1 17152 cofulid 17162 cofurid 17163 catciso 17369 curf2ndf 17499 |
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