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Theorem invf1o 16198
Description: The inverse relation is a bijection from isomorphisms to isomorphisms. This means that every isomorphism 𝐹 ∈ (𝑋𝐼𝑌) has a unique inverse, denoted by ((Inv‘𝐶)‘𝐹). Remark 3.12 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-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)
Assertion
Ref Expression
invf1o (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋))

Proof of Theorem invf1o
StepHypRef Expression
1 invfval.b . . . 4 𝐵 = (Base‘𝐶)
2 invfval.n . . . 4 𝑁 = (Inv‘𝐶)
3 invfval.c . . . 4 (𝜑𝐶 ∈ Cat)
4 invfval.x . . . 4 (𝜑𝑋𝐵)
5 invfval.y . . . 4 (𝜑𝑌𝐵)
6 isoval.n . . . 4 𝐼 = (Iso‘𝐶)
71, 2, 3, 4, 5, 6invf 16197 . . 3 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
8 ffn 5944 . . 3 ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
97, 8syl 17 . 2 (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
101, 2, 3, 5, 4, 6invf 16197 . . . 4 (𝜑 → (𝑌𝑁𝑋):(𝑌𝐼𝑋)⟶(𝑋𝐼𝑌))
11 ffn 5944 . . . 4 ((𝑌𝑁𝑋):(𝑌𝐼𝑋)⟶(𝑋𝐼𝑌) → (𝑌𝑁𝑋) Fn (𝑌𝐼𝑋))
1210, 11syl 17 . . 3 (𝜑 → (𝑌𝑁𝑋) Fn (𝑌𝐼𝑋))
131, 2, 3, 4, 5invsym2 16192 . . . 4 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
1413fneq1d 5881 . . 3 (𝜑 → ((𝑋𝑁𝑌) Fn (𝑌𝐼𝑋) ↔ (𝑌𝑁𝑋) Fn (𝑌𝐼𝑋)))
1512, 14mpbird 245 . 2 (𝜑(𝑋𝑁𝑌) Fn (𝑌𝐼𝑋))
16 dff1o4 6043 . 2 ((𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ (𝑋𝑁𝑌) Fn (𝑌𝐼𝑋)))
179, 15, 16sylanbrc 694 1 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  ccnv 5027   Fn wfn 5785  wf 5786  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  Basecbs 15641  Catccat 16094  Invcinv 16174  Isociso 16175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-cat 16098  df-cid 16099  df-sect 16176  df-inv 16177  df-iso 16178
This theorem is referenced by:  invinv  16199
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