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Theorem invf1o 16252
 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 16251 . . 3 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
8 ffn 5958 . . 3 ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
97, 8syl 17 . 2 (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
101, 2, 3, 5, 4, 6invf 16251 . . . 4 (𝜑 → (𝑌𝑁𝑋):(𝑌𝐼𝑋)⟶(𝑋𝐼𝑌))
11 ffn 5958 . . . 4 ((𝑌𝑁𝑋):(𝑌𝐼𝑋)⟶(𝑋𝐼𝑌) → (𝑌𝑁𝑋) Fn (𝑌𝐼𝑋))
1210, 11syl 17 . . 3 (𝜑 → (𝑌𝑁𝑋) Fn (𝑌𝐼𝑋))
131, 2, 3, 4, 5invsym2 16246 . . . 4 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
1413fneq1d 5895 . . 3 (𝜑 → ((𝑋𝑁𝑌) Fn (𝑌𝐼𝑋) ↔ (𝑌𝑁𝑋) Fn (𝑌𝐼𝑋)))
1512, 14mpbird 246 . 2 (𝜑(𝑋𝑁𝑌) Fn (𝑌𝐼𝑋))
16 dff1o4 6058 . 2 ((𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ (𝑋𝑁𝑌) Fn (𝑌𝐼𝑋)))
179, 15, 16sylanbrc 695 1 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ◡ccnv 5037   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘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:  invinv  16253
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