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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 17710 | . . . 4 ⊢ (𝜑 → Fun (𝑋𝑁𝑌)) |
7 | 6 | funfnd 6569 | . . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)) |
8 | isoval.n | . . . . 5 ⊢ 𝐼 = (Iso‘𝐶) | |
9 | 1, 2, 3, 4, 5, 8 | isoval 17711 | . . . 4 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌)) |
10 | 9 | fneq2d 6633 | . . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))) |
11 | 7, 10 | mpbird 257 | . 2 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌)) |
12 | df-rn 5677 | . . . 4 ⊢ ran (𝑋𝑁𝑌) = dom ◡(𝑋𝑁𝑌) | |
13 | 1, 2, 3, 4, 5 | invsym2 17709 | . . . . . 6 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) |
14 | 13 | dmeqd 5895 | . . . . 5 ⊢ (𝜑 → dom ◡(𝑋𝑁𝑌) = dom (𝑌𝑁𝑋)) |
15 | 1, 2, 3, 5, 4, 8 | isoval 17711 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) = dom (𝑌𝑁𝑋)) |
16 | 14, 15 | eqtr4d 2767 | . . . 4 ⊢ (𝜑 → dom ◡(𝑋𝑁𝑌) = (𝑌𝐼𝑋)) |
17 | 12, 16 | eqtrid 2776 | . . 3 ⊢ (𝜑 → ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋)) |
18 | eqimss 4032 | . . 3 ⊢ (ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋) → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)) | |
19 | 17, 18 | syl 17 | . 2 ⊢ (𝜑 → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)) |
20 | df-f 6537 | . 2 ⊢ ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))) | |
21 | 11, 19, 20 | sylanbrc 582 | 1 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3940 ◡ccnv 5665 dom cdm 5666 ran crn 5667 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 Catccat 17607 Invcinv 17691 Isociso 17692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-cat 17611 df-cid 17612 df-sect 17693 df-inv 17694 df-iso 17695 |
This theorem is referenced by: invf1o 17715 invisoinvl 17736 invcoisoid 17738 isocoinvid 17739 rcaninv 17740 ffthiso 17881 initoeu2lem1 17966 |
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