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Theorem invf 16629
 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 16625 . . . 4 (𝜑 → Fun (𝑋𝑁𝑌))
7 funfn 6079 . . . 4 (Fun (𝑋𝑁𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))
86, 7sylib 208 . . 3 (𝜑 → (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))
9 isoval.n . . . . 5 𝐼 = (Iso‘𝐶)
101, 2, 3, 4, 5, 9isoval 16626 . . . 4 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
1110fneq2d 6143 . . 3 (𝜑 → ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)))
128, 11mpbird 247 . 2 (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
13 df-rn 5277 . . . 4 ran (𝑋𝑁𝑌) = dom (𝑋𝑁𝑌)
141, 2, 3, 4, 5invsym2 16624 . . . . . 6 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
1514dmeqd 5481 . . . . 5 (𝜑 → dom (𝑋𝑁𝑌) = dom (𝑌𝑁𝑋))
161, 2, 3, 5, 4, 9isoval 16626 . . . . 5 (𝜑 → (𝑌𝐼𝑋) = dom (𝑌𝑁𝑋))
1715, 16eqtr4d 2797 . . . 4 (𝜑 → dom (𝑋𝑁𝑌) = (𝑌𝐼𝑋))
1813, 17syl5eq 2806 . . 3 (𝜑 → ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋))
19 eqimss 3798 . . 3 (ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋) → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
2018, 19syl 17 . 2 (𝜑 → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
21 df-f 6053 . 2 ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)))
2212, 20, 21sylanbrc 701 1 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139   ⊆ wss 3715  ◡ccnv 5265  dom cdm 5266  ran crn 5267  Fun wfun 6043   Fn wfn 6044  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813  Basecbs 16059  Catccat 16526  Invcinv 16606  Isociso 16607 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-cat 16530  df-cid 16531  df-sect 16608  df-inv 16609  df-iso 16610 This theorem is referenced by:  invf1o  16630  invisoinvl  16651  invcoisoid  16653  isocoinvid  16654  rcaninv  16655  ffthiso  16790  initoeu2lem1  16865
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