Theorem funciso 17136

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	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
funciso.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
funciso.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
funciso.m	⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼𝑌))

Assertion

Ref	Expression
funciso	⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))

Proof of Theorem funciso

Step	Hyp	Ref	Expression
1		eqid 2819	. 2 ⊢ (Base‘𝐸) = (Base‘𝐸)
2		eqid 2819	. 2 ⊢ (Inv‘𝐸) = (Inv‘𝐸)
3		funciso.f	. . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
4		df-br 5058	. . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
5	3, 4	sylib 220	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
6		funcrcl 17125	. . . 4 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
8	7	simprd 498	. 2 ⊢ (𝜑 → 𝐸 ∈ Cat)
9		funciso.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
10	9, 1, 3	funcf1 17128	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸))
11		funciso.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	10, 11	ffvelrnd 6845	. 2 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸))
13		funciso.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14	10, 13	ffvelrnd 6845	. 2 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸))
15		funciso.t	. 2 ⊢ 𝐽 = (Iso‘𝐸)
16		eqid 2819	. . 3 ⊢ (Inv‘𝐷) = (Inv‘𝐷)
17		funciso.s	. . . 4 ⊢ 𝐼 = (Iso‘𝐷)
18	7	simpld 497	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
19		funciso.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼𝑌))
20	9, 17, 16, 18, 11, 13, 19	invisoinvr 17053	. . 3 ⊢ (𝜑 → 𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀))
21	9, 16, 2, 3, 11, 13, 20	funcinv 17135	. 2 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Inv‘𝐸)(𝐹‘𝑌))((𝑌𝐺𝑋)‘((𝑋(Inv‘𝐷)𝑌)‘𝑀)))
22	1, 2, 8, 12, 14, 15, 21	inviso1 17028	1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1530   ∈ wcel 2107  ⟨cop 4565   class class class wbr 5057  ‘cfv 6348  (class class class)co 7148  Basecbs 16475  Catccat 16927  Invcinv 17007  Isociso 17008   Func cfunc 17116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-map 8400  df-ixp 8454  df-cat 16931  df-cid 16932  df-sect 17009  df-inv 17010  df-iso 17011  df-func 17120

This theorem is referenced by:  ffthiso  17191