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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 17808 | . . . 4 ⊢ (𝜑 → Fun (𝑋𝑁𝑌)) |
| 7 | 6 | funfnd 6597 | . . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)) |
| 8 | isoval.n | . . . . 5 ⊢ 𝐼 = (Iso‘𝐶) | |
| 9 | 1, 2, 3, 4, 5, 8 | isoval 17809 | . . . 4 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌)) |
| 10 | 9 | fneq2d 6662 | . . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))) |
| 11 | 7, 10 | mpbird 257 | . 2 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌)) |
| 12 | df-rn 5696 | . . . 4 ⊢ ran (𝑋𝑁𝑌) = dom ◡(𝑋𝑁𝑌) | |
| 13 | 1, 2, 3, 4, 5 | invsym2 17807 | . . . . . 6 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) |
| 14 | 13 | dmeqd 5916 | . . . . 5 ⊢ (𝜑 → dom ◡(𝑋𝑁𝑌) = dom (𝑌𝑁𝑋)) |
| 15 | 1, 2, 3, 5, 4, 8 | isoval 17809 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) = dom (𝑌𝑁𝑋)) |
| 16 | 14, 15 | eqtr4d 2780 | . . . 4 ⊢ (𝜑 → dom ◡(𝑋𝑁𝑌) = (𝑌𝐼𝑋)) |
| 17 | 12, 16 | eqtrid 2789 | . . 3 ⊢ (𝜑 → ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋)) |
| 18 | eqimss 4042 | . . 3 ⊢ (ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋) → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)) | |
| 19 | 17, 18 | syl 17 | . 2 ⊢ (𝜑 → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)) |
| 20 | df-f 6565 | . 2 ⊢ ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))) | |
| 21 | 11, 19, 20 | sylanbrc 583 | 1 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ◡ccnv 5684 dom cdm 5685 ran crn 5686 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Catccat 17707 Invcinv 17789 Isociso 17790 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-cat 17711 df-cid 17712 df-sect 17791 df-inv 17792 df-iso 17793 |
| This theorem is referenced by: invf1o 17813 invisoinvl 17834 invcoisoid 17836 isocoinvid 17837 rcaninv 17838 ffthiso 17976 initoeu2lem1 18059 upeu2lem 48911 thincciso3 49105 |
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