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Mirrors > Home > MPE Home > Th. List > idfu1 | 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) |
idfu1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
idfu1 | ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋) |
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 | 1, 2, 3 | idfu1st 17868 | . . 3 ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵)) |
5 | 4 | fveq1d 6898 | . 2 ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = (( I ↾ 𝐵)‘𝑋)) |
6 | idfu1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | fvresi 7182 | . . 3 ⊢ (𝑋 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑋) = 𝑋) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → (( I ↾ 𝐵)‘𝑋) = 𝑋) |
9 | 5, 8 | eqtrd 2765 | 1 ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 I cid 5575 ↾ cres 5680 ‘cfv 6549 1st c1st 7992 Basecbs 17183 Catccat 17647 idfunccidfu 17844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-1st 7994 df-idfu 17848 |
This theorem is referenced by: idffth 17925 ressffth 17930 catciso 18103 |
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