Theorem funcinv 16533

Description: The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
funcinv.b	|- B = ( Base ` D )
funcinv.s	|- I = (Inv ` D )
funcinv.t	|- J = (Inv ` E )
funcinv.f	|- ( ph -> F ( D Func E ) G )
funcinv.x	|- ( ph -> X e. B )
funcinv.y	|- ( ph -> Y e. B )
funcinv.m	|- ( ph -> M ( X I Y ) N )

Assertion

Ref	Expression
funcinv	|- ( ph -> ( ( X G Y ) ` M ) ( ( F ` X ) J ( F ` Y ) ) ( ( Y G X ) ` N ) )

Proof of Theorem funcinv

Step	Hyp	Ref	Expression
1		funcinv.b	. . 3 |- B = ( Base ` D )
2		eqid 2622	. . 3 |- (Sect ` D ) = (Sect ` D )
3		eqid 2622	. . 3 |- (Sect ` E ) = (Sect ` E )
4		funcinv.f	. . 3 |- ( ph -> F ( D Func E ) G )
5		funcinv.x	. . 3 |- ( ph -> X e. B )
6		funcinv.y	. . 3 |- ( ph -> Y e. B )
7		funcinv.m	. . . . 5 |- ( ph -> M ( X I Y ) N )
8		funcinv.s	. . . . . 6 |- I = (Inv ` D )
9		df-br 4654	. . . . . . . . 9 |- ( F ( D Func E ) G <-> <. F , G >. e. ( D Func E ) )
10	4, 9	sylib 208	. . . . . . . 8 |- ( ph -> <. F , G >. e. ( D Func E ) )
11		funcrcl 16523	. . . . . . . 8 |- ( <. F , G >. e. ( D Func E ) -> ( D e. Cat /\ E e. Cat ) )
12	10, 11	syl 17	. . . . . . 7 |- ( ph -> ( D e. Cat /\ E e. Cat ) )
13	12	simpld 475	. . . . . 6 |- ( ph -> D e. Cat )
14	1, 8, 13, 5, 6, 2	isinv 16420	. . . . 5 |- ( ph -> ( M ( X I Y ) N <-> ( M ( X (Sect ` D ) Y ) N /\ N ( Y (Sect ` D ) X ) M ) ) )
15	7, 14	mpbid 222	. . . 4 |- ( ph -> ( M ( X (Sect ` D ) Y ) N /\ N ( Y (Sect ` D ) X ) M ) )
16	15	simpld 475	. . 3 |- ( ph -> M ( X (Sect ` D ) Y ) N )
17	1, 2, 3, 4, 5, 6, 16	funcsect 16532	. 2 |- ( ph -> ( ( X G Y ) ` M ) ( ( F ` X ) (Sect ` E ) ( F ` Y ) ) ( ( Y G X ) ` N ) )
18	15	simprd 479	. . 3 |- ( ph -> N ( Y (Sect ` D ) X ) M )
19	1, 2, 3, 4, 6, 5, 18	funcsect 16532	. 2 |- ( ph -> ( ( Y G X ) ` N ) ( ( F ` Y ) (Sect ` E ) ( F ` X ) ) ( ( X G Y ) ` M ) )
20		eqid 2622	. . 3 |- ( Base ` E ) = ( Base ` E )
21		funcinv.t	. . 3 |- J = (Inv ` E )
22	12	simprd 479	. . 3 |- ( ph -> E e. Cat )
23	1, 20, 4	funcf1 16526	. . . 4 |- ( ph -> F : B --> ( Base ` E ) )
24	23, 5	ffvelrnd 6360	. . 3 |- ( ph -> ( F ` X ) e. ( Base ` E ) )
25	23, 6	ffvelrnd 6360	. . 3 |- ( ph -> ( F ` Y ) e. ( Base ` E ) )
26	20, 21, 22, 24, 25, 3	isinv 16420	. 2 |- ( ph -> ( ( ( X G Y ) ` M ) ( ( F ` X ) J ( F ` Y ) ) ( ( Y G X ) ` N ) <-> ( ( ( X G Y ) ` M ) ( ( F ` X ) (Sect ` E ) ( F ` Y ) ) ( ( Y G X ) ` N ) /\ ( ( Y G X ) ` N ) ( ( F ` Y ) (Sect ` E ) ( F ` X ) ) ( ( X G Y ) ` M ) ) ) )
27	17, 19, 26	mpbir2and 957	1 |- ( ph -> ( ( X G Y ) ` M ) ( ( F ` X ) J ( F ` Y ) ) ( ( Y G X ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Catccat 16325  Sectcsect 16404  Invcinv 16405   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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-sect 16407  df-inv 16408  df-func 16518

This theorem is referenced by:  funciso  16534