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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 2733 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
5 | idahom.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | idaval 18008 | . 2 ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ((Id‘𝐶)‘𝑋)⟩) |
7 | idahom.h | . . 3 ⊢ 𝐻 = (Homa‘𝐶) | |
8 | eqid 2733 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
9 | 2, 8, 4, 3, 5 | catidcl 17626 | . . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
10 | 7, 2, 3, 8, 5, 5, 9 | elhomai2 17984 | . 2 ⊢ (𝜑 → ⟨𝑋, 𝑋, ((Id‘𝐶)‘𝑋)⟩ ∈ (𝑋𝐻𝑋)) |
11 | 6, 10 | eqeltrd 2834 | 1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐻𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ⟨cotp 4637 ‘cfv 6544 (class class class)co 7409 Basecbs 17144 Hom chom 17208 Catccat 17608 Idccid 17609 Homachoma 17973 Idacida 18003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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 7365 df-ov 7412 df-cat 17612 df-cid 17613 df-homa 17976 df-ida 18005 |
This theorem is referenced by: idadm 18011 idacd 18012 idaf 18013 arwlid 18022 arwrid 18023 |
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