Theorem funciso 16887

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 2826	. 2 ⊢ (Base‘𝐸) = (Base‘𝐸)
2		eqid 2826	. 2 ⊢ (Inv‘𝐸) = (Inv‘𝐸)
3		funciso.f	. . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
4		df-br 4875	. . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
5	3, 4	sylib 210	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
6		funcrcl 16876	. . . 4 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
8	7	simprd 491	. 2 ⊢ (𝜑 → 𝐸 ∈ Cat)
9		funciso.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
10	9, 1, 3	funcf1 16879	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸))
11		funciso.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	10, 11	ffvelrnd 6610	. 2 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸))
13		funciso.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14	10, 13	ffvelrnd 6610	. 2 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸))
15		funciso.t	. 2 ⊢ 𝐽 = (Iso‘𝐸)
16		eqid 2826	. . 3 ⊢ (Inv‘𝐷) = (Inv‘𝐷)
17		funciso.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼𝑌))
18	7	simpld 490	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
19		funciso.s	. . . . . 6 ⊢ 𝐼 = (Iso‘𝐷)
20	9, 16, 18, 11, 13, 19	isoval 16778	. . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐷)𝑌))
21	17, 20	eleqtrd 2909	. . . 4 ⊢ (𝜑 → 𝑀 ∈ dom (𝑋(Inv‘𝐷)𝑌))
22	9, 16, 18, 11, 13	invfun 16777	. . . . 5 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐷)𝑌))
23		funfvbrb 6580	. . . . 5 ⊢ (Fun (𝑋(Inv‘𝐷)𝑌) → (𝑀 ∈ dom (𝑋(Inv‘𝐷)𝑌) ↔ 𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀)))
24	22, 23	syl 17	. . . 4 ⊢ (𝜑 → (𝑀 ∈ dom (𝑋(Inv‘𝐷)𝑌) ↔ 𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀)))
25	21, 24	mpbid 224	. . 3 ⊢ (𝜑 → 𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀))
26	9, 16, 2, 3, 11, 13, 25	funcinv 16886	. 2 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Inv‘𝐸)(𝐹‘𝑌))((𝑌𝐺𝑋)‘((𝑋(Inv‘𝐷)𝑌)‘𝑀)))
27	1, 2, 8, 12, 14, 15, 26	inviso1 16779	1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ⟨cop 4404   class class class wbr 4874  dom cdm 5343  Fun wfun 6118  ‘cfv 6124  (class class class)co 6906  Basecbs 16223  Catccat 16678  Invcinv 16758  Isociso 16759   Func cfunc 16867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-1st 7429  df-2nd 7430  df-map 8125  df-ixp 8177  df-cat 16682  df-cid 16683  df-sect 16760  df-inv 16761  df-iso 16762  df-func 16871

This theorem is referenced by:  ffthiso  16942