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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 2821 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
5 | idahom.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | idaval 17318 | . 2 ⊢ (𝜑 → (𝐼‘𝑋) = 〈𝑋, 𝑋, ((Id‘𝐶)‘𝑋)〉) |
7 | idahom.h | . . 3 ⊢ 𝐻 = (Homa‘𝐶) | |
8 | eqid 2821 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
9 | 2, 8, 4, 3, 5 | catidcl 16953 | . . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
10 | 7, 2, 3, 8, 5, 5, 9 | elhomai2 17294 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑋, ((Id‘𝐶)‘𝑋)〉 ∈ (𝑋𝐻𝑋)) |
11 | 6, 10 | eqeltrd 2913 | 1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐻𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 〈cotp 4575 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Hom chom 16576 Catccat 16935 Idccid 16936 Homachoma 17283 Idacida 17313 |
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-rmo 3146 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-ot 4576 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-riota 7114 df-ov 7159 df-cat 16939 df-cid 16940 df-homa 17286 df-ida 17315 |
This theorem is referenced by: idadm 17321 idacd 17322 idaf 17323 arwlid 17332 arwrid 17333 |
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