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Theorem funciso 16305
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 2609 . 2 (Base‘𝐸) = (Base‘𝐸)
2 eqid 2609 . 2 (Inv‘𝐸) = (Inv‘𝐸)
3 funciso.f . . . . 5 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
4 df-br 4578 . . . . 5 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
53, 4sylib 206 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
6 funcrcl 16294 . . . 4 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
75, 6syl 17 . . 3 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
87simprd 477 . 2 (𝜑𝐸 ∈ Cat)
9 funciso.b . . . 4 𝐵 = (Base‘𝐷)
109, 1, 3funcf1 16297 . . 3 (𝜑𝐹:𝐵⟶(Base‘𝐸))
11 funciso.x . . 3 (𝜑𝑋𝐵)
1210, 11ffvelrnd 6252 . 2 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
13 funciso.y . . 3 (𝜑𝑌𝐵)
1410, 13ffvelrnd 6252 . 2 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
15 funciso.t . 2 𝐽 = (Iso‘𝐸)
16 eqid 2609 . . 3 (Inv‘𝐷) = (Inv‘𝐷)
17 funciso.m . . . . 5 (𝜑𝑀 ∈ (𝑋𝐼𝑌))
187simpld 473 . . . . . 6 (𝜑𝐷 ∈ Cat)
19 funciso.s . . . . . 6 𝐼 = (Iso‘𝐷)
209, 16, 18, 11, 13, 19isoval 16196 . . . . 5 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐷)𝑌))
2117, 20eleqtrd 2689 . . . 4 (𝜑𝑀 ∈ dom (𝑋(Inv‘𝐷)𝑌))
229, 16, 18, 11, 13invfun 16195 . . . . 5 (𝜑 → Fun (𝑋(Inv‘𝐷)𝑌))
23 funfvbrb 6222 . . . . 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 220 . . 3 (𝜑𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀))
269, 16, 2, 3, 11, 13, 25funcinv 16304 . 2 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Inv‘𝐸)(𝐹𝑌))((𝑌𝐺𝑋)‘((𝑋(Inv‘𝐷)𝑌)‘𝑀)))
271, 2, 8, 12, 14, 15, 26inviso1 16197 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  cop 4130   class class class wbr 4577  dom cdm 5027  Fun wfun 5783  cfv 5789  (class class class)co 6526  Basecbs 15643  Catccat 16096  Invcinv 16176  Isociso 16177   Func cfunc 16285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-1st 7036  df-2nd 7037  df-map 7723  df-ixp 7772  df-cat 16100  df-cid 16101  df-sect 16178  df-inv 16179  df-iso 16180  df-func 16289
This theorem is referenced by:  ffthiso  16360
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