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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 16809 | . . . 4 ⊢ (𝜑 → Fun (𝑋𝑁𝑌)) |
7 | 6 | funfnd 6166 | . . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)) |
8 | isoval.n | . . . . 5 ⊢ 𝐼 = (Iso‘𝐶) | |
9 | 1, 2, 3, 4, 5, 8 | isoval 16810 | . . . 4 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌)) |
10 | 9 | fneq2d 6227 | . . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))) |
11 | 7, 10 | mpbird 249 | . 2 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌)) |
12 | df-rn 5366 | . . . 4 ⊢ ran (𝑋𝑁𝑌) = dom ◡(𝑋𝑁𝑌) | |
13 | 1, 2, 3, 4, 5 | invsym2 16808 | . . . . . 6 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) |
14 | 13 | dmeqd 5571 | . . . . 5 ⊢ (𝜑 → dom ◡(𝑋𝑁𝑌) = dom (𝑌𝑁𝑋)) |
15 | 1, 2, 3, 5, 4, 8 | isoval 16810 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) = dom (𝑌𝑁𝑋)) |
16 | 14, 15 | eqtr4d 2817 | . . . 4 ⊢ (𝜑 → dom ◡(𝑋𝑁𝑌) = (𝑌𝐼𝑋)) |
17 | 12, 16 | syl5eq 2826 | . . 3 ⊢ (𝜑 → ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋)) |
18 | eqimss 3876 | . . 3 ⊢ (ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋) → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)) | |
19 | 17, 18 | syl 17 | . 2 ⊢ (𝜑 → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)) |
20 | df-f 6139 | . 2 ⊢ ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))) | |
21 | 11, 19, 20 | sylanbrc 578 | 1 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 ◡ccnv 5354 dom cdm 5355 ran crn 5356 Fn wfn 6130 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 Catccat 16710 Invcinv 16790 Isociso 16791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-cat 16714 df-cid 16715 df-sect 16792 df-inv 16793 df-iso 16794 |
This theorem is referenced by: invf1o 16814 invisoinvl 16835 invcoisoid 16837 isocoinvid 16838 rcaninv 16839 ffthiso 16974 initoeu2lem1 17049 |
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