Theorem funciso 16534

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	|- ( ph -> F ( D Func E ) G )
funciso.x	|- ( ph -> X e. B )
funciso.y	|- ( ph -> Y e. B )
funciso.m	|- ( ph -> M e. ( X I Y ) )

Assertion

Ref	Expression
funciso	|- ( ph -> ( ( X G Y ) ` M ) e. ( ( F ` X ) J ( F ` Y ) ) )

Proof of Theorem funciso

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( Base ` E ) = ( Base ` E )
2		eqid 2622	. 2 |- (Inv ` E ) = (Inv ` E )
3		funciso.f	. . . . 5 |- ( ph -> F ( D Func E ) G )
4		df-br 4654	. . . . 5 |- ( F ( D Func E ) G <-> <. F , G >. e. ( D Func E ) )
5	3, 4	sylib 208	. . . 4 |- ( ph -> <. F , G >. e. ( D Func E ) )
6		funcrcl 16523	. . . 4 |- ( <. F , G >. e. ( D Func E ) -> ( D e. Cat /\ E e. Cat ) )
7	5, 6	syl 17	. . 3 |- ( ph -> ( D e. Cat /\ E e. Cat ) )
8	7	simprd 479	. 2 |- ( ph -> E e. Cat )
9		funciso.b	. . . 4 |- B = ( Base ` D )
10	9, 1, 3	funcf1 16526	. . 3 |- ( ph -> F : B --> ( Base ` E ) )
11		funciso.x	. . 3 |- ( ph -> X e. B )
12	10, 11	ffvelrnd 6360	. 2 |- ( ph -> ( F ` X ) e. ( Base ` E ) )
13		funciso.y	. . 3 |- ( ph -> Y e. B )
14	10, 13	ffvelrnd 6360	. 2 |- ( ph -> ( F ` Y ) e. ( Base ` E ) )
15		funciso.t	. 2 |- J = ( Iso ` E )
16		eqid 2622	. . 3 |- (Inv ` D ) = (Inv ` D )
17		funciso.m	. . . . 5 |- ( ph -> M e. ( X I Y ) )
18	7	simpld 475	. . . . . 6 |- ( ph -> D e. Cat )
19		funciso.s	. . . . . 6 |- I = ( Iso ` D )
20	9, 16, 18, 11, 13, 19	isoval 16425	. . . . 5 |- ( ph -> ( X I Y ) = dom ( X (Inv ` D ) Y ) )
21	17, 20	eleqtrd 2703	. . . 4 |- ( ph -> M e. dom ( X (Inv ` D ) Y ) )
22	9, 16, 18, 11, 13	invfun 16424	. . . . 5 |- ( ph -> Fun ( X (Inv ` D ) Y ) )
23		funfvbrb 6330	. . . . 5 |- ( Fun ( X (Inv ` D ) Y ) -> ( M e. dom ( X (Inv ` D ) Y ) <-> M ( X (Inv ` D ) Y ) ( ( X (Inv ` D ) Y ) ` M ) ) )
24	22, 23	syl 17	. . . 4 |- ( ph -> ( M e. dom ( X (Inv ` D ) Y ) <-> M ( X (Inv ` D ) Y ) ( ( X (Inv ` D ) Y ) ` M ) ) )
25	21, 24	mpbid 222	. . 3 |- ( ph -> M ( X (Inv ` D ) Y ) ( ( X (Inv ` D ) Y ) ` M ) )
26	9, 16, 2, 3, 11, 13, 25	funcinv 16533	. 2 |- ( ph -> ( ( X G Y ) ` M ) ( ( F ` X ) (Inv ` E ) ( F ` Y ) ) ( ( Y G X ) ` ( ( X (Inv ` D ) Y ) ` M ) ) )
27	1, 2, 8, 12, 14, 15, 26	inviso1 16426	1 |- ( ph -> ( ( X G Y ) ` M ) e. ( ( F ` X ) J ( F ` Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   dom cdm 5114   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Catccat 16325  Invcinv 16405   Iso ciso 16406   Func cfunc 16514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-cat 16329  df-cid 16330  df-sect 16407  df-inv 16408  df-iso 16409  df-func 16518

This theorem is referenced by:  ffthiso  16589