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Mirrors > Home > MPE Home > Th. List > idahom | Structured version Visualization version GIF version |
Description: Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
idafval.i | ⊢ 𝐼 = (Ida‘𝐶) |
idafval.b | ⊢ 𝐵 = (Base‘𝐶) |
idafval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
idahom.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
idahom.h | ⊢ 𝐻 = (Homa‘𝐶) |
Ref | Expression |
---|---|
idahom | ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐻𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idafval.i | . . 3 ⊢ 𝐼 = (Ida‘𝐶) | |
2 | idafval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
3 | idafval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | eqid 2730 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
5 | idahom.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | idaval 18014 | . 2 ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ((Id‘𝐶)‘𝑋)⟩) |
7 | idahom.h | . . 3 ⊢ 𝐻 = (Homa‘𝐶) | |
8 | eqid 2730 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
9 | 2, 8, 4, 3, 5 | catidcl 17632 | . . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
10 | 7, 2, 3, 8, 5, 5, 9 | elhomai2 17990 | . 2 ⊢ (𝜑 → ⟨𝑋, 𝑋, ((Id‘𝐶)‘𝑋)⟩ ∈ (𝑋𝐻𝑋)) |
11 | 6, 10 | eqeltrd 2831 | 1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐻𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 ⟨cotp 4637 ‘cfv 6544 (class class class)co 7413 Basecbs 17150 Hom chom 17214 Catccat 17614 Idccid 17615 Homachoma 17979 Idacida 18009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-ot 4638 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-cat 17618 df-cid 17619 df-homa 17982 df-ida 18011 |
This theorem is referenced by: idadm 18017 idacd 18018 idaf 18019 arwlid 18028 arwrid 18029 |
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