MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  inviso2 Structured version   Visualization version   GIF version

Theorem inviso2 17812
Description: If 𝐺 is an inverse to 𝐹, then 𝐺 is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
invfval.b 𝐵 = (Base‘𝐶)
invfval.n 𝑁 = (Inv‘𝐶)
invfval.c (𝜑𝐶 ∈ Cat)
invfval.x (𝜑𝑋𝐵)
invfval.y (𝜑𝑌𝐵)
isoval.n 𝐼 = (Iso‘𝐶)
inviso1.1 (𝜑𝐹(𝑋𝑁𝑌)𝐺)
Assertion
Ref Expression
inviso2 (𝜑𝐺 ∈ (𝑌𝐼𝑋))

Proof of Theorem inviso2
StepHypRef Expression
1 invfval.b . 2 𝐵 = (Base‘𝐶)
2 invfval.n . 2 𝑁 = (Inv‘𝐶)
3 invfval.c . 2 (𝜑𝐶 ∈ Cat)
4 invfval.y . 2 (𝜑𝑌𝐵)
5 invfval.x . 2 (𝜑𝑋𝐵)
6 isoval.n . 2 𝐼 = (Iso‘𝐶)
7 inviso1.1 . . 3 (𝜑𝐹(𝑋𝑁𝑌)𝐺)
81, 2, 3, 5, 4invsym 17807 . . 3 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺𝐺(𝑌𝑁𝑋)𝐹))
97, 8mpbid 232 . 2 (𝜑𝐺(𝑌𝑁𝑋)𝐹)
101, 2, 3, 4, 5, 6, 9inviso1 17811 1 (𝜑𝐺 ∈ (𝑌𝐼𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107   class class class wbr 5142  cfv 6560  (class class class)co 7432  Basecbs 17248  Catccat 17708  Invcinv 17790  Isociso 17791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-cat 17712  df-cid 17713  df-sect 17792  df-inv 17793  df-iso 17794
This theorem is referenced by:  yonffthlem  18328
  Copyright terms: Public domain W3C validator