Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > idiso | Structured version Visualization version GIF version |
Description: The identity is an isomorphism. Example 3.13 of [Adamek] p. 28. (Contributed by AV, 8-Apr-2020.) |
Ref | Expression |
---|---|
invid.b | ⊢ 𝐵 = (Base‘𝐶) |
invid.i | ⊢ 𝐼 = (Id‘𝐶) |
invid.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
invid.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
idiso | ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋(Iso‘𝐶)𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invid.b | . 2 ⊢ 𝐵 = (Base‘𝐶) | |
2 | eqid 2738 | . 2 ⊢ (Inv‘𝐶) = (Inv‘𝐶) | |
3 | invid.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | invid.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | eqid 2738 | . 2 ⊢ (Iso‘𝐶) = (Iso‘𝐶) | |
6 | invid.i | . . 3 ⊢ 𝐼 = (Id‘𝐶) | |
7 | 1, 6, 3, 4 | invid 17509 | . 2 ⊢ (𝜑 → (𝐼‘𝑋)(𝑋(Inv‘𝐶)𝑋)(𝐼‘𝑋)) |
8 | 1, 2, 3, 4, 4, 5, 7 | inviso1 17488 | 1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋(Iso‘𝐶)𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ‘cfv 6426 (class class class)co 7267 Basecbs 16922 Catccat 17383 Idccid 17384 Invcinv 17467 Isociso 17468 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-1st 7820 df-2nd 7821 df-cat 17387 df-cid 17388 df-sect 17469 df-inv 17470 df-iso 17471 |
This theorem is referenced by: invisoinvl 17512 cicref 17523 |
Copyright terms: Public domain | W3C validator |