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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 17752 | . . . 4 ⊢ (𝜑 → Fun (𝑋𝑁𝑌)) |
7 | funrel 6573 | . . . 4 ⊢ (Fun (𝑋𝑁𝑌) → Rel (𝑋𝑁𝑌)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → Rel (𝑋𝑁𝑌)) |
9 | inviso1.1 | . . 3 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) | |
10 | releldm 5948 | . . 3 ⊢ ((Rel (𝑋𝑁𝑌) ∧ 𝐹(𝑋𝑁𝑌)𝐺) → 𝐹 ∈ dom (𝑋𝑁𝑌)) | |
11 | 8, 9, 10 | syl2anc 582 | . 2 ⊢ (𝜑 → 𝐹 ∈ dom (𝑋𝑁𝑌)) |
12 | isoval.n | . . 3 ⊢ 𝐼 = (Iso‘𝐶) | |
13 | 1, 2, 3, 4, 5, 12 | isoval 17753 | . 2 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌)) |
14 | 11, 13 | eleqtrrd 2831 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 class class class wbr 5150 dom cdm 5680 Rel wrel 5685 Fun wfun 6545 ‘cfv 6551 (class class class)co 7424 Basecbs 17185 Catccat 17649 Invcinv 17733 Isociso 17734 |
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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 7997 df-2nd 7998 df-cat 17653 df-cid 17654 df-sect 17735 df-inv 17736 df-iso 17737 |
This theorem is referenced by: inviso2 17755 isoco 17765 idiso 17776 funciso 17865 ffthiso 17923 fuciso 17972 initoeu1 18005 termoeu1 18012 catciso 18105 yoneda 18280 |
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