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Theorem invf 17038
 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 17034 . . . 4 (𝜑 → Fun (𝑋𝑁𝑌))
76funfnd 6374 . . 3 (𝜑 → (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))
8 isoval.n . . . . 5 𝐼 = (Iso‘𝐶)
91, 2, 3, 4, 5, 8isoval 17035 . . . 4 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
109fneq2d 6435 . . 3 (𝜑 → ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)))
117, 10mpbird 260 . 2 (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
12 df-rn 5553 . . . 4 ran (𝑋𝑁𝑌) = dom (𝑋𝑁𝑌)
131, 2, 3, 4, 5invsym2 17033 . . . . . 6 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
1413dmeqd 5761 . . . . 5 (𝜑 → dom (𝑋𝑁𝑌) = dom (𝑌𝑁𝑋))
151, 2, 3, 5, 4, 8isoval 17035 . . . . 5 (𝜑 → (𝑌𝐼𝑋) = dom (𝑌𝑁𝑋))
1614, 15eqtr4d 2862 . . . 4 (𝜑 → dom (𝑋𝑁𝑌) = (𝑌𝐼𝑋))
1712, 16syl5eq 2871 . . 3 (𝜑 → ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋))
18 eqimss 4009 . . 3 (ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋) → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
1917, 18syl 17 . 2 (𝜑 → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
20 df-f 6347 . 2 ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)))
2111, 19, 20sylanbrc 586 1 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115   ⊆ wss 3919  ◡ccnv 5541  dom cdm 5542  ran crn 5543   Fn wfn 6338  ⟶wf 6339  ‘cfv 6343  (class class class)co 7149  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 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-cat 16939  df-cid 16940  df-sect 17017  df-inv 17018  df-iso 17019 This theorem is referenced by:  invf1o  17039  invisoinvl  17060  invcoisoid  17062  isocoinvid  17063  rcaninv  17064  ffthiso  17199  initoeu2lem1  17274
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