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Mirrors > Home > MPE Home > Th. List > invf | Structured version Visualization version GIF version |
Description: The inverse relation is a function from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
invfval.b | ⊢ 𝐵 = (Base‘𝐶) |
invfval.n | ⊢ 𝑁 = (Inv‘𝐶) |
invfval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
invfval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
invfval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
isoval.n | ⊢ 𝐼 = (Iso‘𝐶) |
Ref | Expression |
---|---|
invf | ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
2 | invfval.n | . . . . 5 ⊢ 𝑁 = (Inv‘𝐶) | |
3 | invfval.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | invfval.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | invfval.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | invfun 17028 | . . . 4 ⊢ (𝜑 → Fun (𝑋𝑁𝑌)) |
7 | 6 | funfnd 6380 | . . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)) |
8 | isoval.n | . . . . 5 ⊢ 𝐼 = (Iso‘𝐶) | |
9 | 1, 2, 3, 4, 5, 8 | isoval 17029 | . . . 4 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌)) |
10 | 9 | fneq2d 6441 | . . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))) |
11 | 7, 10 | mpbird 259 | . 2 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌)) |
12 | df-rn 5560 | . . . 4 ⊢ ran (𝑋𝑁𝑌) = dom ◡(𝑋𝑁𝑌) | |
13 | 1, 2, 3, 4, 5 | invsym2 17027 | . . . . . 6 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) |
14 | 13 | dmeqd 5768 | . . . . 5 ⊢ (𝜑 → dom ◡(𝑋𝑁𝑌) = dom (𝑌𝑁𝑋)) |
15 | 1, 2, 3, 5, 4, 8 | isoval 17029 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) = dom (𝑌𝑁𝑋)) |
16 | 14, 15 | eqtr4d 2859 | . . . 4 ⊢ (𝜑 → dom ◡(𝑋𝑁𝑌) = (𝑌𝐼𝑋)) |
17 | 12, 16 | syl5eq 2868 | . . 3 ⊢ (𝜑 → ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋)) |
18 | eqimss 4022 | . . 3 ⊢ (ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋) → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)) | |
19 | 17, 18 | syl 17 | . 2 ⊢ (𝜑 → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)) |
20 | df-f 6353 | . 2 ⊢ ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))) | |
21 | 11, 19, 20 | sylanbrc 585 | 1 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ◡ccnv 5548 dom cdm 5549 ran crn 5550 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Catccat 16929 Invcinv 17009 Isociso 17010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-cat 16933 df-cid 16934 df-sect 17011 df-inv 17012 df-iso 17013 |
This theorem is referenced by: invf1o 17033 invisoinvl 17054 invcoisoid 17056 isocoinvid 17057 rcaninv 17058 ffthiso 17193 initoeu2lem1 17268 |
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