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Theorem funciso 16740
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 2770 . 2 (Base‘𝐸) = (Base‘𝐸)
2 eqid 2770 . 2 (Inv‘𝐸) = (Inv‘𝐸)
3 funciso.f . . . . 5 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
4 df-br 4785 . . . . 5 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
53, 4sylib 208 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
6 funcrcl 16729 . . . 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 16732 . . 3 (𝜑𝐹:𝐵⟶(Base‘𝐸))
11 funciso.x . . 3 (𝜑𝑋𝐵)
1210, 11ffvelrnd 6503 . 2 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
13 funciso.y . . 3 (𝜑𝑌𝐵)
1410, 13ffvelrnd 6503 . 2 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
15 funciso.t . 2 𝐽 = (Iso‘𝐸)
16 eqid 2770 . . 3 (Inv‘𝐷) = (Inv‘𝐷)
17 funciso.m . . . . 5 (𝜑𝑀 ∈ (𝑋𝐼𝑌))
187simpld 476 . . . . . 6 (𝜑𝐷 ∈ Cat)
19 funciso.s . . . . . 6 𝐼 = (Iso‘𝐷)
209, 16, 18, 11, 13, 19isoval 16631 . . . . 5 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐷)𝑌))
2117, 20eleqtrd 2851 . . . 4 (𝜑𝑀 ∈ dom (𝑋(Inv‘𝐷)𝑌))
229, 16, 18, 11, 13invfun 16630 . . . . 5 (𝜑 → Fun (𝑋(Inv‘𝐷)𝑌))
23 funfvbrb 6473 . . . . 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 222 . . 3 (𝜑𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀))
269, 16, 2, 3, 11, 13, 25funcinv 16739 . 2 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Inv‘𝐸)(𝐹𝑌))((𝑌𝐺𝑋)‘((𝑋(Inv‘𝐷)𝑌)‘𝑀)))
271, 2, 8, 12, 14, 15, 26inviso1 16632 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  cop 4320   class class class wbr 4784  dom cdm 5249  Fun wfun 6025  cfv 6031  (class class class)co 6792  Basecbs 16063  Catccat 16531  Invcinv 16611  Isociso 16612   Func cfunc 16720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-map 8010  df-ixp 8062  df-cat 16535  df-cid 16536  df-sect 16613  df-inv 16614  df-iso 16615  df-func 16724
This theorem is referenced by:  ffthiso  16795
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