Theorem xpcomco 6323

Description: Composition with the bijection of xpcomf1o 6322 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)

Hypotheses

Ref	Expression
xpcomf1o.1	|- F = ( x e. ( A X. B ) |-> U. `' { x } )
xpcomco.1	|- G = ( y e. B , z e. A |-> C )

Assertion

Ref	Expression
xpcomco	|- ( G o. F ) = ( z e. A , y e. B |-> C )

Distinct variable groups:   x, y, z, A   x, B, y, z   y, F, z

Allowed substitution hints:   C( x, y, z)   F( x)   G( x, y, z)

Proof of Theorem xpcomco

Dummy variables v u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpcomf1o.1	. . . . . . . . . 10 |- F = ( x e. ( A X. B ) |-> U. `' { x } )
2	1	xpcomf1o 6322	. . . . . . . . 9 |- F : ( A X. B ) -1-1-onto-> ( B X. A )
3		f1ofun 5148	. . . . . . . . 9 |- ( F : ( A X. B ) -1-1-onto-> ( B X. A ) -> Fun F )
4		funbrfv2b 5239	. . . . . . . . 9 |- ( Fun F -> ( u F w <-> ( u e. dom F /\ ( F ` u ) = w ) ) )
5	2, 3, 4	mp2b 8	. . . . . . . 8 |- ( u F w <-> ( u e. dom F /\ ( F ` u ) = w ) )
6		ancom 262	. . . . . . . 8 |- ( ( u e. dom F /\ ( F ` u ) = w ) <-> ( ( F ` u ) = w /\ u e. dom F ) )
7		eqcom 2083	. . . . . . . . 9 |- ( ( F ` u ) = w <-> w = ( F ` u ) )
8		f1odm 5150	. . . . . . . . . . 11 |- ( F : ( A X. B ) -1-1-onto-> ( B X. A ) -> dom F = ( A X. B ) )
9	2, 8	ax-mp 7	. . . . . . . . . 10 |- dom F = ( A X. B )
10	9	eleq2i 2145	. . . . . . . . 9 |- ( u e. dom F <-> u e. ( A X. B ) )
11	7, 10	anbi12i 447	. . . . . . . 8 |- ( ( ( F ` u ) = w /\ u e. dom F ) <-> ( w = ( F ` u ) /\ u e. ( A X. B ) ) )
12	5, 6, 11	3bitri 204	. . . . . . 7 |- ( u F w <-> ( w = ( F ` u ) /\ u e. ( A X. B ) ) )
13	12	anbi1i 445	. . . . . 6 |- ( ( u F w /\ w G v ) <-> ( ( w = ( F ` u ) /\ u e. ( A X. B ) ) /\ w G v ) )
14		anass 393	. . . . . 6 |- ( ( ( w = ( F ` u ) /\ u e. ( A X. B ) ) /\ w G v ) <-> ( w = ( F ` u ) /\ ( u e. ( A X. B ) /\ w G v ) ) )
15	13, 14	bitri 182	. . . . 5 |- ( ( u F w /\ w G v ) <-> ( w = ( F ` u ) /\ ( u e. ( A X. B ) /\ w G v ) ) )
16	15	exbii 1536	. . . 4 |- ( E. w ( u F w /\ w G v ) <-> E. w ( w = ( F ` u ) /\ ( u e. ( A X. B ) /\ w G v ) ) )
17		vex 2604	. . . . . . 7 |- u e. _V
18	1	mptfvex 5277	. . . . . . 7 |- ( ( A. x U. `' { x } e. _V /\ u e. _V ) -> ( F ` u ) e. _V )
19	17, 18	mpan2 415	. . . . . 6 |- ( A. x U. `' { x } e. _V -> ( F ` u ) e. _V )
20		vex 2604	. . . . . . . . 9 |- x e. _V
21	20	snex 3957	. . . . . . . 8 |- { x } e. _V
22	21	cnvex 4876	. . . . . . 7 |- `' { x } e. _V
23	22	uniex 4192	. . . . . 6 |- U. `' { x } e. _V
24	19, 23	mpg 1380	. . . . 5 |- ( F ` u ) e. _V
25		breq1 3788	. . . . . 6 |- ( w = ( F ` u ) -> ( w G v <-> ( F ` u ) G v ) )
26	25	anbi2d 451	. . . . 5 |- ( w = ( F ` u ) -> ( ( u e. ( A X. B ) /\ w G v ) <-> ( u e. ( A X. B ) /\ ( F ` u ) G v ) ) )
27	24, 26	ceqsexv 2638	. . . 4 |- ( E. w ( w = ( F ` u ) /\ ( u e. ( A X. B ) /\ w G v ) ) <-> ( u e. ( A X. B ) /\ ( F ` u ) G v ) )
28		elxp 4380	. . . . . 6 |- ( u e. ( A X. B ) <-> E. z E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) )
29	28	anbi1i 445	. . . . 5 |- ( ( u e. ( A X. B ) /\ ( F ` u ) G v ) <-> ( E. z E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) )
30		nfcv 2219	. . . . . . 7 |- F/_ z ( F ` u )
31		xpcomco.1	. . . . . . . 8 |- G = ( y e. B , z e. A |-> C )
32		nfmpt22 5592	. . . . . . . 8 |- F/_ z ( y e. B , z e. A |-> C )
33	31, 32	nfcxfr 2216	. . . . . . 7 |- F/_ z G
34		nfcv 2219	. . . . . . 7 |- F/_ z v
35	30, 33, 34	nfbr 3829	. . . . . 6 |- F/ z ( F ` u ) G v
36	35	19.41 1616	. . . . 5 |- ( E. z ( E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> ( E. z E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) )
37		nfcv 2219	. . . . . . . . 9 |- F/_ y ( F ` u )
38		nfmpt21 5591	. . . . . . . . . 10 |- F/_ y ( y e. B , z e. A |-> C )
39	31, 38	nfcxfr 2216	. . . . . . . . 9 |- F/_ y G
40		nfcv 2219	. . . . . . . . 9 |- F/_ y v
41	37, 39, 40	nfbr 3829	. . . . . . . 8 |- F/ y ( F ` u ) G v
42	41	19.41 1616	. . . . . . 7 |- ( E. y ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> ( E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) )
43		anass 393	. . . . . . . . 9 |- ( ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ ( F ` u ) G v ) ) )
44		fveq2 5198	. . . . . . . . . . . . . 14 |- ( u = <. z , y >. -> ( F ` u ) = ( F ` <. z , y >. ) )
45		opelxpi 4394	. . . . . . . . . . . . . . 15 |- ( ( z e. A /\ y e. B ) -> <. z , y >. e. ( A X. B ) )
46		sneq 3409	. . . . . . . . . . . . . . . . . . 19 |- ( x = <. z , y >. -> { x } = { <. z , y >. } )
47	46	cnveqd 4529	. . . . . . . . . . . . . . . . . 18 |- ( x = <. z , y >. -> `' { x } = `' { <. z , y >. } )
48	47	unieqd 3612	. . . . . . . . . . . . . . . . 17 |- ( x = <. z , y >. -> U. `' { x } = U. `' { <. z , y >. } )
49		vex 2604	. . . . . . . . . . . . . . . . . 18 |- z e. _V
50		vex 2604	. . . . . . . . . . . . . . . . . 18 |- y e. _V
51		opswapg 4827	. . . . . . . . . . . . . . . . . 18 |- ( ( z e. _V /\ y e. _V ) -> U. `' { <. z , y >. } = <. y , z >. )
52	49, 50, 51	mp2an 416	. . . . . . . . . . . . . . . . 17 |- U. `' { <. z , y >. } = <. y , z >.
53	48, 52	syl6eq 2129	. . . . . . . . . . . . . . . 16 |- ( x = <. z , y >. -> U. `' { x } = <. y , z >. )
54	50, 49	opex 3984	. . . . . . . . . . . . . . . 16 |- <. y , z >. e. _V
55	53, 1, 54	fvmpt 5270	. . . . . . . . . . . . . . 15 |- ( <. z , y >. e. ( A X. B ) -> ( F ` <. z , y >. ) = <. y , z >. )
56	45, 55	syl 14	. . . . . . . . . . . . . 14 |- ( ( z e. A /\ y e. B ) -> ( F ` <. z , y >. ) = <. y , z >. )
57	44, 56	sylan9eq 2133	. . . . . . . . . . . . 13 |- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( F ` u ) = <. y , z >. )
58	57	breq1d 3795	. . . . . . . . . . . 12 |- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( ( F ` u ) G v <-> <. y , z >. G v ) )
59		df-br 3786	. . . . . . . . . . . . . . . 16 |- ( <. y , z >. G v <-> <. <. y , z >. , v >. e. G )
60		df-mpt2 5537	. . . . . . . . . . . . . . . . . 18 |- ( y e. B , z e. A |-> C ) = { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) }
61	31, 60	eqtri 2101	. . . . . . . . . . . . . . . . 17 |- G = { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) }
62	61	eleq2i 2145	. . . . . . . . . . . . . . . 16 |- ( <. <. y , z >. , v >. e. G <-> <. <. y , z >. , v >. e. { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) } )
63		oprabid 5557	. . . . . . . . . . . . . . . 16 |- ( <. <. y , z >. , v >. e. { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) } <-> ( ( y e. B /\ z e. A ) /\ v = C ) )
64	59, 62, 63	3bitri 204	. . . . . . . . . . . . . . 15 |- ( <. y , z >. G v <-> ( ( y e. B /\ z e. A ) /\ v = C ) )
65	64	baib 861	. . . . . . . . . . . . . 14 |- ( ( y e. B /\ z e. A ) -> ( <. y , z >. G v <-> v = C ) )
66	65	ancoms 264	. . . . . . . . . . . . 13 |- ( ( z e. A /\ y e. B ) -> ( <. y , z >. G v <-> v = C ) )
67	66	adantl 271	. . . . . . . . . . . 12 |- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( <. y , z >. G v <-> v = C ) )
68	58, 67	bitrd 186	. . . . . . . . . . 11 |- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( ( F ` u ) G v <-> v = C ) )
69	68	pm5.32da 439	. . . . . . . . . 10 |- ( u = <. z , y >. -> ( ( ( z e. A /\ y e. B ) /\ ( F ` u ) G v ) <-> ( ( z e. A /\ y e. B ) /\ v = C ) ) )
70	69	pm5.32i 441	. . . . . . . . 9 |- ( ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ ( F ` u ) G v ) ) <-> ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) )
71	43, 70	bitri 182	. . . . . . . 8 |- ( ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) )
72	71	exbii 1536	. . . . . . 7 |- ( E. y ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) )
73	42, 72	bitr3i 184	. . . . . 6 |- ( ( E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) )
74	73	exbii 1536	. . . . 5 |- ( E. z ( E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> E. z E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) )
75	29, 36, 74	3bitr2i 206	. . . 4 |- ( ( u e. ( A X. B ) /\ ( F ` u ) G v ) <-> E. z E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) )
76	16, 27, 75	3bitri 204	. . 3 |- ( E. w ( u F w /\ w G v ) <-> E. z E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) )
77	76	opabbii 3845	. 2 |- { <. u , v >. | E. w ( u F w /\ w G v ) } = { <. u , v >. | E. z E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) }
78		df-co 4372	. 2 |- ( G o. F ) = { <. u , v >. | E. w ( u F w /\ w G v ) }
79		df-mpt2 5537	. . 3 |- ( z e. A , y e. B |-> C ) = { <. <. z , y >. , v >. | ( ( z e. A /\ y e. B ) /\ v = C ) }
80		dfoprab2 5572	. . 3 |- { <. <. z , y >. , v >. | ( ( z e. A /\ y e. B ) /\ v = C ) } = { <. u , v >. | E. z E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) }
81	79, 80	eqtri 2101	. 2 |- ( z e. A , y e. B |-> C ) = { <. u , v >. | E. z E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) }
82	77, 78, 81	3eqtr4i 2111	1 |- ( G o. F ) = ( z e. A , y e. B |-> C )

Syntax hints:   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   {csn 3398   <.cop 3401   U.cuni 3601   class class class wbr 3785   {copab 3838   |-> cmpt 3839   X. cxp 4361   `'ccnv 4362   dom cdm 4363   o. ccom 4367   Fun wfun 4916   -1-1-onto->wf1o 4921   ` cfv 4922   {coprab 5533   |-> cmpt2 5534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788

This theorem is referenced by: (None)