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Mirrors > Home > MPE Home > Th. List > inviso1 | Structured version Visualization version GIF version |
Description: If 𝐺 is an inverse to 𝐹, then 𝐹 is an isomorphism. (Contributed by Mario Carneiro, 3-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) |
inviso1.1 | ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) |
Ref | Expression |
---|---|
inviso1 | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) |
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 | funrel 5821 | . . . 4 ⊢ (Fun (𝑋𝑁𝑌) → Rel (𝑋𝑁𝑌)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → Rel (𝑋𝑁𝑌)) |
9 | inviso1.1 | . . 3 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) | |
10 | releldm 5279 | . . 3 ⊢ ((Rel (𝑋𝑁𝑌) ∧ 𝐹(𝑋𝑁𝑌)𝐺) → 𝐹 ∈ dom (𝑋𝑁𝑌)) | |
11 | 8, 9, 10 | syl2anc 691 | . 2 ⊢ (𝜑 → 𝐹 ∈ dom (𝑋𝑁𝑌)) |
12 | isoval.n | . . 3 ⊢ 𝐼 = (Iso‘𝐶) | |
13 | 1, 2, 3, 4, 5, 12 | isoval 16248 | . 2 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌)) |
14 | 11, 13 | eleqtrrd 2691 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 dom cdm 5038 Rel wrel 5043 Fun wfun 5798 ‘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: inviso2 16250 isoco 16260 idiso 16271 cicref 16284 funciso 16357 ffthiso 16412 fuciso 16458 initoeu1 16484 termoeu1 16491 catciso 16580 yoneda 16746 |
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