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Theorem invf 17812
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 17808 . . . 4 (𝜑 → Fun (𝑋𝑁𝑌))
76funfnd 6597 . . 3 (𝜑 → (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))
8 isoval.n . . . . 5 𝐼 = (Iso‘𝐶)
91, 2, 3, 4, 5, 8isoval 17809 . . . 4 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
109fneq2d 6662 . . 3 (𝜑 → ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)))
117, 10mpbird 257 . 2 (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
12 df-rn 5696 . . . 4 ran (𝑋𝑁𝑌) = dom (𝑋𝑁𝑌)
131, 2, 3, 4, 5invsym2 17807 . . . . . 6 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
1413dmeqd 5916 . . . . 5 (𝜑 → dom (𝑋𝑁𝑌) = dom (𝑌𝑁𝑋))
151, 2, 3, 5, 4, 8isoval 17809 . . . . 5 (𝜑 → (𝑌𝐼𝑋) = dom (𝑌𝑁𝑋))
1614, 15eqtr4d 2780 . . . 4 (𝜑 → dom (𝑋𝑁𝑌) = (𝑌𝐼𝑋))
1712, 16eqtrid 2789 . . 3 (𝜑 → ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋))
18 eqimss 4042 . . 3 (ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋) → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
1917, 18syl 17 . 2 (𝜑 → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
20 df-f 6565 . 2 ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)))
2111, 19, 20sylanbrc 583 1 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wss 3951  ccnv 5684  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  Catccat 17707  Invcinv 17789  Isociso 17790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-cat 17711  df-cid 17712  df-sect 17791  df-inv 17792  df-iso 17793
This theorem is referenced by:  invf1o  17813  invisoinvl  17834  invcoisoid  17836  isocoinvid  17837  rcaninv  17838  ffthiso  17976  initoeu2lem1  18059  upeu2lem  48911  thincciso3  49105
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