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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 16247 | . . . 4 ⊢ (𝜑 → Fun (𝑋𝑁𝑌)) |
7 | funfn 5833 | . . . 4 ⊢ (Fun (𝑋𝑁𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)) | |
8 | 6, 7 | sylib 207 | . . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)) |
9 | isoval.n | . . . . 5 ⊢ 𝐼 = (Iso‘𝐶) | |
10 | 1, 2, 3, 4, 5, 9 | isoval 16248 | . . . 4 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌)) |
11 | 10 | fneq2d 5896 | . . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))) |
12 | 8, 11 | mpbird 246 | . 2 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌)) |
13 | df-rn 5049 | . . . 4 ⊢ ran (𝑋𝑁𝑌) = dom ◡(𝑋𝑁𝑌) | |
14 | 1, 2, 3, 4, 5 | invsym2 16246 | . . . . . 6 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) |
15 | 14 | dmeqd 5248 | . . . . 5 ⊢ (𝜑 → dom ◡(𝑋𝑁𝑌) = dom (𝑌𝑁𝑋)) |
16 | 1, 2, 3, 5, 4, 9 | isoval 16248 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) = dom (𝑌𝑁𝑋)) |
17 | 15, 16 | eqtr4d 2647 | . . . 4 ⊢ (𝜑 → dom ◡(𝑋𝑁𝑌) = (𝑌𝐼𝑋)) |
18 | 13, 17 | syl5eq 2656 | . . 3 ⊢ (𝜑 → ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋)) |
19 | eqimss 3620 | . . 3 ⊢ (ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋) → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)) | |
20 | 18, 19 | syl 17 | . 2 ⊢ (𝜑 → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)) |
21 | df-f 5808 | . 2 ⊢ ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))) | |
22 | 12, 20, 21 | sylanbrc 695 | 1 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ◡ccnv 5037 dom cdm 5038 ran crn 5039 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Catccat 16148 Invcinv 16228 Isociso 16229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-cat 16152 df-cid 16153 df-sect 16230 df-inv 16231 df-iso 16232 |
This theorem is referenced by: invf1o 16252 invisoinvl 16273 invcoisoid 16275 isocoinvid 16276 rcaninv 16277 ffthiso 16412 initoeu2lem1 16487 |
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