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Theorem funciso 16357
Description: The image of an isomorphism under a functor is an isomorphism. Proposition 3.21 of [Adamek] p. 32. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
funciso.b 𝐵 = (Base‘𝐷)
funciso.s 𝐼 = (Iso‘𝐷)
funciso.t 𝐽 = (Iso‘𝐸)
funciso.f (𝜑𝐹(𝐷 Func 𝐸)𝐺)
funciso.x (𝜑𝑋𝐵)
funciso.y (𝜑𝑌𝐵)
funciso.m (𝜑𝑀 ∈ (𝑋𝐼𝑌))
Assertion
Ref Expression
funciso (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))

Proof of Theorem funciso
StepHypRef Expression
1 eqid 2610 . 2 (Base‘𝐸) = (Base‘𝐸)
2 eqid 2610 . 2 (Inv‘𝐸) = (Inv‘𝐸)
3 funciso.f . . . . 5 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
4 df-br 4584 . . . . 5 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
53, 4sylib 207 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
6 funcrcl 16346 . . . 4 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
75, 6syl 17 . . 3 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
87simprd 478 . 2 (𝜑𝐸 ∈ Cat)
9 funciso.b . . . 4 𝐵 = (Base‘𝐷)
109, 1, 3funcf1 16349 . . 3 (𝜑𝐹:𝐵⟶(Base‘𝐸))
11 funciso.x . . 3 (𝜑𝑋𝐵)
1210, 11ffvelrnd 6268 . 2 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
13 funciso.y . . 3 (𝜑𝑌𝐵)
1410, 13ffvelrnd 6268 . 2 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
15 funciso.t . 2 𝐽 = (Iso‘𝐸)
16 eqid 2610 . . 3 (Inv‘𝐷) = (Inv‘𝐷)
17 funciso.m . . . . 5 (𝜑𝑀 ∈ (𝑋𝐼𝑌))
187simpld 474 . . . . . 6 (𝜑𝐷 ∈ Cat)
19 funciso.s . . . . . 6 𝐼 = (Iso‘𝐷)
209, 16, 18, 11, 13, 19isoval 16248 . . . . 5 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐷)𝑌))
2117, 20eleqtrd 2690 . . . 4 (𝜑𝑀 ∈ dom (𝑋(Inv‘𝐷)𝑌))
229, 16, 18, 11, 13invfun 16247 . . . . 5 (𝜑 → Fun (𝑋(Inv‘𝐷)𝑌))
23 funfvbrb 6238 . . . . 5 (Fun (𝑋(Inv‘𝐷)𝑌) → (𝑀 ∈ dom (𝑋(Inv‘𝐷)𝑌) ↔ 𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀)))
2422, 23syl 17 . . . 4 (𝜑 → (𝑀 ∈ dom (𝑋(Inv‘𝐷)𝑌) ↔ 𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀)))
2521, 24mpbid 221 . . 3 (𝜑𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀))
269, 16, 2, 3, 11, 13, 25funcinv 16356 . 2 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Inv‘𝐸)(𝐹𝑌))((𝑌𝐺𝑋)‘((𝑋(Inv‘𝐷)𝑌)‘𝑀)))
271, 2, 8, 12, 14, 15, 26inviso1 16249 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  cop 4131   class class class wbr 4583  dom cdm 5038  Fun wfun 5798  cfv 5804  (class class class)co 6549  Basecbs 15695  Catccat 16148  Invcinv 16228  Isociso 16229   Func cfunc 16337
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-map 7746  df-ixp 7795  df-cat 16152  df-cid 16153  df-sect 16230  df-inv 16231  df-iso 16232  df-func 16341
This theorem is referenced by:  ffthiso  16412
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