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Theorem invf 16251
Description: The inverse relation is a function from isomorphisms to isomorphisms. (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
invf (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))

Proof of Theorem invf
StepHypRef Expression
1 invfval.b . . . . 5 𝐵 = (Base‘𝐶)
2 invfval.n . . . . 5 𝑁 = (Inv‘𝐶)
3 invfval.c . . . . 5 (𝜑𝐶 ∈ Cat)
4 invfval.x . . . . 5 (𝜑𝑋𝐵)
5 invfval.y . . . . 5 (𝜑𝑌𝐵)
61, 2, 3, 4, 5invfun 16247 . . . 4 (𝜑 → Fun (𝑋𝑁𝑌))
7 funfn 5833 . . . 4 (Fun (𝑋𝑁𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))
86, 7sylib 207 . . 3 (𝜑 → (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))
9 isoval.n . . . . 5 𝐼 = (Iso‘𝐶)
101, 2, 3, 4, 5, 9isoval 16248 . . . 4 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
1110fneq2d 5896 . . 3 (𝜑 → ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)))
128, 11mpbird 246 . 2 (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
13 df-rn 5049 . . . 4 ran (𝑋𝑁𝑌) = dom (𝑋𝑁𝑌)
141, 2, 3, 4, 5invsym2 16246 . . . . . 6 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
1514dmeqd 5248 . . . . 5 (𝜑 → dom (𝑋𝑁𝑌) = dom (𝑌𝑁𝑋))
161, 2, 3, 5, 4, 9isoval 16248 . . . . 5 (𝜑 → (𝑌𝐼𝑋) = dom (𝑌𝑁𝑋))
1715, 16eqtr4d 2647 . . . 4 (𝜑 → dom (𝑋𝑁𝑌) = (𝑌𝐼𝑋))
1813, 17syl5eq 2656 . . 3 (𝜑 → ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋))
19 eqimss 3620 . . 3 (ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋) → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
2018, 19syl 17 . 2 (𝜑 → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
21 df-f 5808 . 2 ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)))
2212, 20, 21sylanbrc 695 1 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  wss 3540  ccnv 5037  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  wf 5800  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:  invf1o  16252  invisoinvl  16273  invcoisoid  16275  isocoinvid  16276  rcaninv  16277  ffthiso  16412  initoeu2lem1  16487
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