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Theorem funciso 17123
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 2820 . 2 (Base‘𝐸) = (Base‘𝐸)
2 eqid 2820 . 2 (Inv‘𝐸) = (Inv‘𝐸)
3 funciso.f . . . . 5 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
4 df-br 5043 . . . . 5 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
53, 4sylib 220 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
6 funcrcl 17112 . . . 4 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
75, 6syl 17 . . 3 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
87simprd 498 . 2 (𝜑𝐸 ∈ Cat)
9 funciso.b . . . 4 𝐵 = (Base‘𝐷)
109, 1, 3funcf1 17115 . . 3 (𝜑𝐹:𝐵⟶(Base‘𝐸))
11 funciso.x . . 3 (𝜑𝑋𝐵)
1210, 11ffvelrnd 6828 . 2 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
13 funciso.y . . 3 (𝜑𝑌𝐵)
1410, 13ffvelrnd 6828 . 2 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
15 funciso.t . 2 𝐽 = (Iso‘𝐸)
16 eqid 2820 . . 3 (Inv‘𝐷) = (Inv‘𝐷)
17 funciso.s . . . 4 𝐼 = (Iso‘𝐷)
187simpld 497 . . . 4 (𝜑𝐷 ∈ Cat)
19 funciso.m . . . 4 (𝜑𝑀 ∈ (𝑋𝐼𝑌))
209, 17, 16, 18, 11, 13, 19invisoinvr 17040 . . 3 (𝜑𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀))
219, 16, 2, 3, 11, 13, 20funcinv 17122 . 2 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Inv‘𝐸)(𝐹𝑌))((𝑌𝐺𝑋)‘((𝑋(Inv‘𝐷)𝑌)‘𝑀)))
221, 2, 8, 12, 14, 15, 21inviso1 17015 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  cop 4549   class class class wbr 5042  cfv 6331  (class class class)co 7133  Basecbs 16462  Catccat 16914  Invcinv 16994  Isociso 16995   Func cfunc 17103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-1st 7667  df-2nd 7668  df-map 8386  df-ixp 8440  df-cat 16918  df-cid 16919  df-sect 16996  df-inv 16997  df-iso 16998  df-func 17107
This theorem is referenced by:  ffthiso  17178
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