Theorem funciso 17776

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 2731	. 2 ⊢ (Base‘𝐸) = (Base‘𝐸)
2		eqid 2731	. 2 ⊢ (Inv‘𝐸) = (Inv‘𝐸)
3		funciso.f	. . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
4		df-br 5087	. . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
5	3, 4	sylib 218	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
6		funcrcl 17765	. . . 4 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
8	7	simprd 495	. 2 ⊢ (𝜑 → 𝐸 ∈ Cat)
9		funciso.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
10	9, 1, 3	funcf1 17768	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸))
11		funciso.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	10, 11	ffvelcdmd 7013	. 2 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸))
13		funciso.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14	10, 13	ffvelcdmd 7013	. 2 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸))
15		funciso.t	. 2 ⊢ 𝐽 = (Iso‘𝐸)
16		eqid 2731	. . 3 ⊢ (Inv‘𝐷) = (Inv‘𝐷)
17		funciso.s	. . . 4 ⊢ 𝐼 = (Iso‘𝐷)
18	7	simpld 494	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
19		funciso.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼𝑌))
20	9, 17, 16, 18, 11, 13, 19	invisoinvr 17693	. . 3 ⊢ (𝜑 → 𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀))
21	9, 16, 2, 3, 11, 13, 20	funcinv 17775	. 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 17668	1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⟨cop 4577   class class class wbr 5086  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  Catccat 17565  Invcinv 17647  Isociso 17648   Func cfunc 17756

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-map 8747  df-ixp 8817  df-cat 17569  df-cid 17570  df-sect 17649  df-inv 17650  df-iso 17651  df-func 17760

This theorem is referenced by:  ffthiso  17833