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Theorem funciso 17820
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 2733 . 2 (Base‘𝐸) = (Base‘𝐸)
2 eqid 2733 . 2 (Inv‘𝐸) = (Inv‘𝐸)
3 funciso.f . . . . 5 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
4 df-br 5148 . . . . 5 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
53, 4sylib 217 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
6 funcrcl 17809 . . . 4 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
75, 6syl 17 . . 3 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
87simprd 497 . 2 (𝜑𝐸 ∈ Cat)
9 funciso.b . . . 4 𝐵 = (Base‘𝐷)
109, 1, 3funcf1 17812 . . 3 (𝜑𝐹:𝐵⟶(Base‘𝐸))
11 funciso.x . . 3 (𝜑𝑋𝐵)
1210, 11ffvelcdmd 7083 . 2 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
13 funciso.y . . 3 (𝜑𝑌𝐵)
1410, 13ffvelcdmd 7083 . 2 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
15 funciso.t . 2 𝐽 = (Iso‘𝐸)
16 eqid 2733 . . 3 (Inv‘𝐷) = (Inv‘𝐷)
17 funciso.s . . . 4 𝐼 = (Iso‘𝐷)
187simpld 496 . . . 4 (𝜑𝐷 ∈ Cat)
19 funciso.m . . . 4 (𝜑𝑀 ∈ (𝑋𝐼𝑌))
209, 17, 16, 18, 11, 13, 19invisoinvr 17734 . . 3 (𝜑𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀))
219, 16, 2, 3, 11, 13, 20funcinv 17819 . 2 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Inv‘𝐸)(𝐹𝑌))((𝑌𝐺𝑋)‘((𝑋(Inv‘𝐷)𝑌)‘𝑀)))
221, 2, 8, 12, 14, 15, 21inviso1 17709 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cop 4633   class class class wbr 5147  cfv 6540  (class class class)co 7404  Basecbs 17140  Catccat 17604  Invcinv 17688  Isociso 17689   Func cfunc 17800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7970  df-2nd 7971  df-map 8818  df-ixp 8888  df-cat 17608  df-cid 17609  df-sect 17690  df-inv 17691  df-iso 17692  df-func 17804
This theorem is referenced by:  ffthiso  17876
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