Theorem xpcomco 6936

Description: Composition with the bijection of xpcomf1o 6935 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 6935	. . . . . . . . 9 |- F : ( A X. B ) -1-1-onto-> ( B X. A )
3		f1ofun 5536	. . . . . . . . 9 |- ( F : ( A X. B ) -1-1-onto-> ( B X. A ) -> Fun F )
4		funbrfv2b 5636	. . . . . . . . 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 2208	. . . . . . . . 9 |- ( ( F ` u ) = w <-> w = ( F ` u ) )
8		f1odm 5538	. . . . . . . . . . 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 2273	. . . . . . . . 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 1629	. . . 4 |- ( E. w ( u F w /\ w G v ) <-> E. w ( w = ( F ` u ) /\ ( u e. ( A X. B ) /\ w G v ) ) )
17		vex 2776	. . . . . . 7 |- u e. _V
18	1	mptfvex 5678	. . . . . . 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 2776	. . . . . . . . 9 |- x e. _V
21	20	snex 4237	. . . . . . . 8 |- { x } e. _V
22	21	cnvex 5230	. . . . . . 7 |- `' { x } e. _V
23	22	uniex 4492	. . . . . 6 |- U. `' { x } e. _V
24	19, 23	mpg 1475	. . . . 5 |- ( F ` u ) e. _V
25		breq1 4054	. . . . . 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 2813	. . . 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 4700	. . . . . 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 2349	. . . . . . 7 |- F/_ z ( F ` u )
31		xpcomco.1	. . . . . . . 8 |- G = ( y e. B , z e. A |-> C )
32		nfmpo2 6026	. . . . . . . 8 |- F/_ z ( y e. B , z e. A |-> C )
33	31, 32	nfcxfr 2346	. . . . . . 7 |- F/_ z G
34		nfcv 2349	. . . . . . 7 |- F/_ z v
35	30, 33, 34	nfbr 4098	. . . . . 6 |- F/ z ( F ` u ) G v
36	35	19.41 1710	. . . . 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 2349	. . . . . . . . 9 |- F/_ y ( F ` u )
38		nfmpo1 6025	. . . . . . . . . 10 |- F/_ y ( y e. B , z e. A |-> C )
39	31, 38	nfcxfr 2346	. . . . . . . . 9 |- F/_ y G
40		nfcv 2349	. . . . . . . . 9 |- F/_ y v
41	37, 39, 40	nfbr 4098	. . . . . . . 8 |- F/ y ( F ` u ) G v
42	41	19.41 1710	. . . . . . 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 5589	. . . . . . . . . . . . . 14 |- ( u = <. z , y >. -> ( F ` u ) = ( F ` <. z , y >. ) )
45		opelxpi 4715	. . . . . . . . . . . . . . 15 |- ( ( z e. A /\ y e. B ) -> <. z , y >. e. ( A X. B ) )
46		sneq 3649	. . . . . . . . . . . . . . . . . . 19 |- ( x = <. z , y >. -> { x } = { <. z , y >. } )
47	46	cnveqd 4862	. . . . . . . . . . . . . . . . . 18 |- ( x = <. z , y >. -> `' { x } = `' { <. z , y >. } )
48	47	unieqd 3867	. . . . . . . . . . . . . . . . 17 |- ( x = <. z , y >. -> U. `' { x } = U. `' { <. z , y >. } )
49		vex 2776	. . . . . . . . . . . . . . . . . 18 |- z e. _V
50		vex 2776	. . . . . . . . . . . . . . . . . 18 |- y e. _V
51		opswapg 5178	. . . . . . . . . . . . . . . . . 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 2255	. . . . . . . . . . . . . . . 16 |- ( x = <. z , y >. -> U. `' { x } = <. y , z >. )
54	50, 49	opex 4281	. . . . . . . . . . . . . . . 16 |- <. y , z >. e. _V
55	53, 1, 54	fvmpt 5669	. . . . . . . . . . . . . . 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 2259	. . . . . . . . . . . . 13 |- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( F ` u ) = <. y , z >. )
58	57	breq1d 4061	. . . . . . . . . . . 12 |- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( ( F ` u ) G v <-> <. y , z >. G v ) )
59		df-br 4052	. . . . . . . . . . . . . . . 16 |- ( <. y , z >. G v <-> <. <. y , z >. , v >. e. G )
60		df-mpo 5962	. . . . . . . . . . . . . . . . . 18 |- ( y e. B , z e. A |-> C ) = { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) }
61	31, 60	eqtri 2227	. . . . . . . . . . . . . . . . 17 |- G = { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) }
62	61	eleq2i 2273	. . . . . . . . . . . . . . . 16 |- ( <. <. y , z >. , v >. e. G <-> <. <. y , z >. , v >. e. { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) } )
63		oprabid 5989	. . . . . . . . . . . . . . . 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 921	. . . . . . . . . . . . . 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 1629	. . . . . . 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 1629	. . . . 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 4119	. 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 4692	. 2 |- ( G o. F ) = { <. u , v >. | E. w ( u F w /\ w G v ) }
79		df-mpo 5962	. . 3 |- ( z e. A , y e. B |-> C ) = { <. <. z , y >. , v >. | ( ( z e. A /\ y e. B ) /\ v = C ) }
80		dfoprab2 6005	. . 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 2227	. 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 2237	1 |- ( G o. F ) = ( z e. A , y e. B |-> C )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1516   e. wcel 2177   _Vcvv 2773   {csn 3638   <.cop 3641   U.cuni 3856   class class class wbr 4051   {copab 4112   |-> cmpt 4113   X. cxp 4681   `'ccnv 4682   dom cdm 4683   o. ccom 4687   Fun wfun 5274   -1-1-onto->wf1o 5279   ` cfv 5280   {coprab 5958   e. cmpo 5959

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240

This theorem is referenced by: (None)