Theorem xpcomco 6820

Description: Composition with the bijection of xpcomf1o 6819 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 6819	. . . . . . . . 9 |- F : ( A X. B ) -1-1-onto-> ( B X. A )
3		f1ofun 5459	. . . . . . . . 9 |- ( F : ( A X. B ) -1-1-onto-> ( B X. A ) -> Fun F )
4		funbrfv2b 5556	. . . . . . . . 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 266	. . . . . . . 8 |- ( ( u e. dom F /\ ( F ` u ) = w ) <-> ( ( F ` u ) = w /\ u e. dom F ) )
7		eqcom 2179	. . . . . . . . 9 |- ( ( F ` u ) = w <-> w = ( F ` u ) )
8		f1odm 5461	. . . . . . . . . . 11 |- ( F : ( A X. B ) -1-1-onto-> ( B X. A ) -> dom F = ( A X. B ) )
9	2, 8	ax-mp 5	. . . . . . . . . 10 |- dom F = ( A X. B )
10	9	eleq2i 2244	. . . . . . . . 9 |- ( u e. dom F <-> u e. ( A X. B ) )
11	7, 10	anbi12i 460	. . . . . . . 8 |- ( ( ( F ` u ) = w /\ u e. dom F ) <-> ( w = ( F ` u ) /\ u e. ( A X. B ) ) )
12	5, 6, 11	3bitri 206	. . . . . . 7 |- ( u F w <-> ( w = ( F ` u ) /\ u e. ( A X. B ) ) )
13	12	anbi1i 458	. . . . . 6 |- ( ( u F w /\ w G v ) <-> ( ( w = ( F ` u ) /\ u e. ( A X. B ) ) /\ w G v ) )
14		anass 401	. . . . . 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 184	. . . . 5 |- ( ( u F w /\ w G v ) <-> ( w = ( F ` u ) /\ ( u e. ( A X. B ) /\ w G v ) ) )
16	15	exbii 1605	. . . 4 |- ( E. w ( u F w /\ w G v ) <-> E. w ( w = ( F ` u ) /\ ( u e. ( A X. B ) /\ w G v ) ) )
17		vex 2740	. . . . . . 7 |- u e. _V
18	1	mptfvex 5597	. . . . . . 7 |- ( ( A. x U. `' { x } e. _V /\ u e. _V ) -> ( F ` u ) e. _V )
19	17, 18	mpan2 425	. . . . . 6 |- ( A. x U. `' { x } e. _V -> ( F ` u ) e. _V )
20		vex 2740	. . . . . . . . 9 |- x e. _V
21	20	snex 4182	. . . . . . . 8 |- { x } e. _V
22	21	cnvex 5163	. . . . . . 7 |- `' { x } e. _V
23	22	uniex 4434	. . . . . 6 |- U. `' { x } e. _V
24	19, 23	mpg 1451	. . . . 5 |- ( F ` u ) e. _V
25		breq1 4003	. . . . . 6 |- ( w = ( F ` u ) -> ( w G v <-> ( F ` u ) G v ) )
26	25	anbi2d 464	. . . . 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 2776	. . . 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 4640	. . . . . 6 |- ( u e. ( A X. B ) <-> E. z E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) )
29	28	anbi1i 458	. . . . 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 2319	. . . . . . 7 |- F/_ z ( F ` u )
31		xpcomco.1	. . . . . . . 8 |- G = ( y e. B , z e. A |-> C )
32		nfmpo2 5937	. . . . . . . 8 |- F/_ z ( y e. B , z e. A |-> C )
33	31, 32	nfcxfr 2316	. . . . . . 7 |- F/_ z G
34		nfcv 2319	. . . . . . 7 |- F/_ z v
35	30, 33, 34	nfbr 4046	. . . . . 6 |- F/ z ( F ` u ) G v
36	35	19.41 1686	. . . . 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 2319	. . . . . . . . 9 |- F/_ y ( F ` u )
38		nfmpo1 5936	. . . . . . . . . 10 |- F/_ y ( y e. B , z e. A |-> C )
39	31, 38	nfcxfr 2316	. . . . . . . . 9 |- F/_ y G
40		nfcv 2319	. . . . . . . . 9 |- F/_ y v
41	37, 39, 40	nfbr 4046	. . . . . . . 8 |- F/ y ( F ` u ) G v
42	41	19.41 1686	. . . . . . 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 401	. . . . . . . . 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 5511	. . . . . . . . . . . . . 14 |- ( u = <. z , y >. -> ( F ` u ) = ( F ` <. z , y >. ) )
45		opelxpi 4655	. . . . . . . . . . . . . . 15 |- ( ( z e. A /\ y e. B ) -> <. z , y >. e. ( A X. B ) )
46		sneq 3602	. . . . . . . . . . . . . . . . . . 19 |- ( x = <. z , y >. -> { x } = { <. z , y >. } )
47	46	cnveqd 4799	. . . . . . . . . . . . . . . . . 18 |- ( x = <. z , y >. -> `' { x } = `' { <. z , y >. } )
48	47	unieqd 3818	. . . . . . . . . . . . . . . . 17 |- ( x = <. z , y >. -> U. `' { x } = U. `' { <. z , y >. } )
49		vex 2740	. . . . . . . . . . . . . . . . . 18 |- z e. _V
50		vex 2740	. . . . . . . . . . . . . . . . . 18 |- y e. _V
51		opswapg 5111	. . . . . . . . . . . . . . . . . 18 |- ( ( z e. _V /\ y e. _V ) -> U. `' { <. z , y >. } = <. y , z >. )
52	49, 50, 51	mp2an 426	. . . . . . . . . . . . . . . . 17 |- U. `' { <. z , y >. } = <. y , z >.
53	48, 52	eqtrdi 2226	. . . . . . . . . . . . . . . 16 |- ( x = <. z , y >. -> U. `' { x } = <. y , z >. )
54	50, 49	opex 4226	. . . . . . . . . . . . . . . 16 |- <. y , z >. e. _V
55	53, 1, 54	fvmpt 5589	. . . . . . . . . . . . . . 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 2230	. . . . . . . . . . . . 13 |- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( F ` u ) = <. y , z >. )
58	57	breq1d 4010	. . . . . . . . . . . 12 |- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( ( F ` u ) G v <-> <. y , z >. G v ) )
59		df-br 4001	. . . . . . . . . . . . . . . 16 |- ( <. y , z >. G v <-> <. <. y , z >. , v >. e. G )
60		df-mpo 5874	. . . . . . . . . . . . . . . . . 18 |- ( y e. B , z e. A |-> C ) = { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) }
61	31, 60	eqtri 2198	. . . . . . . . . . . . . . . . 17 |- G = { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) }
62	61	eleq2i 2244	. . . . . . . . . . . . . . . 16 |- ( <. <. y , z >. , v >. e. G <-> <. <. y , z >. , v >. e. { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) } )
63		oprabid 5901	. . . . . . . . . . . . . . . 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 206	. . . . . . . . . . . . . . 15 |- ( <. y , z >. G v <-> ( ( y e. B /\ z e. A ) /\ v = C ) )
65	64	baib 919	. . . . . . . . . . . . . 14 |- ( ( y e. B /\ z e. A ) -> ( <. y , z >. G v <-> v = C ) )
66	65	ancoms 268	. . . . . . . . . . . . 13 |- ( ( z e. A /\ y e. B ) -> ( <. y , z >. G v <-> v = C ) )
67	66	adantl 277	. . . . . . . . . . . 12 |- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( <. y , z >. G v <-> v = C ) )
68	58, 67	bitrd 188	. . . . . . . . . . 11 |- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( ( F ` u ) G v <-> v = C ) )
69	68	pm5.32da 452	. . . . . . . . . 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 454	. . . . . . . . 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 184	. . . . . . . 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 1605	. . . . . . 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 186	. . . . . 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 1605	. . . . 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 208	. . . 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 206	. . 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 4067	. 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 4632	. 2 |- ( G o. F ) = { <. u , v >. | E. w ( u F w /\ w G v ) }
79		df-mpo 5874	. . 3 |- ( z e. A , y e. B |-> C ) = { <. <. z , y >. , v >. | ( ( z e. A /\ y e. B ) /\ v = C ) }
80		dfoprab2 5916	. . 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 2198	. 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 2208	1 |- ( G o. F ) = ( z e. A , y e. B |-> C )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2737   {csn 3591   <.cop 3594   U.cuni 3807   class class class wbr 4000   {copab 4060   |-> cmpt 4061   X. cxp 4621   `'ccnv 4622   dom cdm 4623   o. ccom 4627   Fun wfun 5206   -1-1-onto->wf1o 5211   ` cfv 5212   {coprab 5870   e. cmpo 5871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136

This theorem is referenced by: (None)