Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > invisoinvr | Structured version Visualization version GIF version |
Description: The inverse of an isomorphism is invers to the isomorphism. (Contributed by AV, 9-Apr-2020.) |
Ref | Expression |
---|---|
invisoinv.b | ⊢ 𝐵 = (Base‘𝐶) |
invisoinv.i | ⊢ 𝐼 = (Iso‘𝐶) |
invisoinv.n | ⊢ 𝑁 = (Inv‘𝐶) |
invisoinv.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
invisoinv.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
invisoinv.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
invisoinv.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) |
Ref | Expression |
---|---|
invisoinvr | ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invisoinv.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | invisoinv.i | . . 3 ⊢ 𝐼 = (Iso‘𝐶) | |
3 | invisoinv.n | . . 3 ⊢ 𝑁 = (Inv‘𝐶) | |
4 | invisoinv.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | invisoinv.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | invisoinv.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | invisoinv.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | invisoinvl 17267 | . 2 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹) |
9 | 1, 3, 4, 5, 6 | invsym 17239 | . 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)) |
10 | 8, 9 | mpbird 260 | 1 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 Catccat 17139 Invcinv 17222 Isociso 17223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-cat 17143 df-cid 17144 df-sect 17224 df-inv 17225 df-iso 17226 |
This theorem is referenced by: invcoisoid 17269 funciso 17352 fuciso 17456 |
Copyright terms: Public domain | W3C validator |