Theorem inviso2 17037

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	⊢ (𝜑 → 𝐶 ∈ Cat)
invfval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invfval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
isoval.n	⊢ 𝐼 = (Iso‘𝐶)
inviso1.1	⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺)

Assertion

Ref	Expression
inviso2	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑋))

Proof of Theorem inviso2

Step	Hyp	Ref	Expression
1		invfval.b	. 2 ⊢ 𝐵 = (Base‘𝐶)
2		invfval.n	. 2 ⊢ 𝑁 = (Inv‘𝐶)
3		invfval.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		invfval.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
5		invfval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		isoval.n	. 2 ⊢ 𝐼 = (Iso‘𝐶)
7		inviso1.1	. . 3 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺)
8	1, 2, 3, 5, 4	invsym 17032	. . 3 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐺(𝑌𝑁𝑋)𝐹))
9	7, 8	mpbid 234	. 2 ⊢ (𝜑 → 𝐺(𝑌𝑁𝑋)𝐹)
10	1, 2, 3, 4, 5, 6, 9	inviso1 17036	1 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑋))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  Catccat 16935  Invcinv 17015  Isociso 17016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-cat 16939  df-cid 16940  df-sect 17017  df-inv 17018  df-iso 17019

This theorem is referenced by:  yonffthlem  17532