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Theorem invf 17714
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 17710 . . . 4 (𝜑 → Fun (𝑋𝑁𝑌))
76funfnd 6569 . . 3 (𝜑 → (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))
8 isoval.n . . . . 5 𝐼 = (Iso‘𝐶)
91, 2, 3, 4, 5, 8isoval 17711 . . . 4 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
109fneq2d 6633 . . 3 (𝜑 → ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)))
117, 10mpbird 257 . 2 (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
12 df-rn 5677 . . . 4 ran (𝑋𝑁𝑌) = dom (𝑋𝑁𝑌)
131, 2, 3, 4, 5invsym2 17709 . . . . . 6 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
1413dmeqd 5895 . . . . 5 (𝜑 → dom (𝑋𝑁𝑌) = dom (𝑌𝑁𝑋))
151, 2, 3, 5, 4, 8isoval 17711 . . . . 5 (𝜑 → (𝑌𝐼𝑋) = dom (𝑌𝑁𝑋))
1614, 15eqtr4d 2767 . . . 4 (𝜑 → dom (𝑋𝑁𝑌) = (𝑌𝐼𝑋))
1712, 16eqtrid 2776 . . 3 (𝜑 → ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋))
18 eqimss 4032 . . 3 (ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋) → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
1917, 18syl 17 . 2 (𝜑 → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
20 df-f 6537 . 2 ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)))
2111, 19, 20sylanbrc 582 1 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wss 3940  ccnv 5665  dom cdm 5666  ran crn 5667   Fn wfn 6528  wf 6529  cfv 6533  (class class class)co 7401  Basecbs 17143  Catccat 17607  Invcinv 17691  Isociso 17692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-cat 17611  df-cid 17612  df-sect 17693  df-inv 17694  df-iso 17695
This theorem is referenced by:  invf1o  17715  invisoinvl  17736  invcoisoid  17738  isocoinvid  17739  rcaninv  17740  ffthiso  17881  initoeu2lem1  17966
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