Theorem opeliunxp 4413

Description: Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)

Assertion

Ref	Expression
opeliunxp	|- ( <. x , C >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ C e. B ) )

Proof of Theorem opeliunxp

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2610	. 2 |- ( <. x , C >. e. U_ x e. A ( { x } X. B ) -> <. x , C >. e. _V )
2		opexg 3983	. 2 |- ( ( x e. A /\ C e. B ) -> <. x , C >. e. _V )
3		df-rex 2354	. . . . . 6 |- ( E. x e. A y e. ( { x } X. B ) <-> E. x ( x e. A /\ y e. ( { x } X. B ) ) )
4		nfv 1461	. . . . . . 7 |- F/ z ( x e. A /\ y e. ( { x } X. B ) )
5		nfs1v 1856	. . . . . . . 8 |- F/ x [ z / x ] x e. A
6		nfcv 2219	. . . . . . . . . 10 |- F/_ x { z }
7		nfcsb1v 2938	. . . . . . . . . 10 |- F/_ x [_ z / x ]_ B
8	6, 7	nfxp 4389	. . . . . . . . 9 |- F/_ x ( { z } X. [_ z / x ]_ B )
9	8	nfcri 2213	. . . . . . . 8 |- F/ x y e. ( { z } X. [_ z / x ]_ B )
10	5, 9	nfan 1497	. . . . . . 7 |- F/ x ( [ z / x ] x e. A /\ y e. ( { z } X. [_ z / x ]_ B ) )
11		sbequ12 1694	. . . . . . . 8 |- ( x = z -> ( x e. A <-> [ z / x ] x e. A ) )
12		sneq 3409	. . . . . . . . . 10 |- ( x = z -> { x } = { z } )
13		csbeq1a 2916	. . . . . . . . . 10 |- ( x = z -> B = [_ z / x ]_ B )
14	12, 13	xpeq12d 4388	. . . . . . . . 9 |- ( x = z -> ( { x } X. B ) = ( { z } X. [_ z / x ]_ B ) )
15	14	eleq2d 2148	. . . . . . . 8 |- ( x = z -> ( y e. ( { x } X. B ) <-> y e. ( { z } X. [_ z / x ]_ B ) ) )
16	11, 15	anbi12d 456	. . . . . . 7 |- ( x = z -> ( ( x e. A /\ y e. ( { x } X. B ) ) <-> ( [ z / x ] x e. A /\ y e. ( { z } X. [_ z / x ]_ B ) ) ) )
17	4, 10, 16	cbvex 1679	. . . . . 6 |- ( E. x ( x e. A /\ y e. ( { x } X. B ) ) <-> E. z ( [ z / x ] x e. A /\ y e. ( { z } X. [_ z / x ]_ B ) ) )
18	3, 17	bitri 182	. . . . 5 |- ( E. x e. A y e. ( { x } X. B ) <-> E. z ( [ z / x ] x e. A /\ y e. ( { z } X. [_ z / x ]_ B ) ) )
19		eleq1 2141	. . . . . . 7 |- ( y = <. x , C >. -> ( y e. ( { z } X. [_ z / x ]_ B ) <-> <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) )
20	19	anbi2d 451	. . . . . 6 |- ( y = <. x , C >. -> ( ( [ z / x ] x e. A /\ y e. ( { z } X. [_ z / x ]_ B ) ) <-> ( [ z / x ] x e. A /\ <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) ) )
21	20	exbidv 1746	. . . . 5 |- ( y = <. x , C >. -> ( E. z ( [ z / x ] x e. A /\ y e. ( { z } X. [_ z / x ]_ B ) ) <-> E. z ( [ z / x ] x e. A /\ <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) ) )
22	18, 21	syl5bb 190	. . . 4 |- ( y = <. x , C >. -> ( E. x e. A y e. ( { x } X. B ) <-> E. z ( [ z / x ] x e. A /\ <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) ) )
23		df-iun 3680	. . . 4 |- U_ x e. A ( { x } X. B ) = { y | E. x e. A y e. ( { x } X. B ) }
24	22, 23	elab2g 2740	. . 3 |- ( <. x , C >. e. _V -> ( <. x , C >. e. U_ x e. A ( { x } X. B ) <-> E. z ( [ z / x ] x e. A /\ <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) ) )
25		opelxp 4392	. . . . . . 7 |- ( <. x , C >. e. ( { z } X. [_ z / x ]_ B ) <-> ( x e. { z } /\ C e. [_ z / x ]_ B ) )
26	25	anbi2i 444	. . . . . 6 |- ( ( [ z / x ] x e. A /\ <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) <-> ( [ z / x ] x e. A /\ ( x e. { z } /\ C e. [_ z / x ]_ B ) ) )
27		an12 525	. . . . . 6 |- ( ( [ z / x ] x e. A /\ ( x e. { z } /\ C e. [_ z / x ]_ B ) ) <-> ( x e. { z } /\ ( [ z / x ] x e. A /\ C e. [_ z / x ]_ B ) ) )
28		velsn 3415	. . . . . . . 8 |- ( x e. { z } <-> x = z )
29		equcom 1633	. . . . . . . 8 |- ( x = z <-> z = x )
30	28, 29	bitri 182	. . . . . . 7 |- ( x e. { z } <-> z = x )
31	30	anbi1i 445	. . . . . 6 |- ( ( x e. { z } /\ ( [ z / x ] x e. A /\ C e. [_ z / x ]_ B ) ) <-> ( z = x /\ ( [ z / x ] x e. A /\ C e. [_ z / x ]_ B ) ) )
32	26, 27, 31	3bitri 204	. . . . 5 |- ( ( [ z / x ] x e. A /\ <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) <-> ( z = x /\ ( [ z / x ] x e. A /\ C e. [_ z / x ]_ B ) ) )
33	32	exbii 1536	. . . 4 |- ( E. z ( [ z / x ] x e. A /\ <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) <-> E. z ( z = x /\ ( [ z / x ] x e. A /\ C e. [_ z / x ]_ B ) ) )
34		vex 2604	. . . . 5 |- x e. _V
35		sbequ12r 1695	. . . . . 6 |- ( z = x -> ( [ z / x ] x e. A <-> x e. A ) )
36	13	equcoms 1634	. . . . . . . 8 |- ( z = x -> B = [_ z / x ]_ B )
37	36	eqcomd 2086	. . . . . . 7 |- ( z = x -> [_ z / x ]_ B = B )
38	37	eleq2d 2148	. . . . . 6 |- ( z = x -> ( C e. [_ z / x ]_ B <-> C e. B ) )
39	35, 38	anbi12d 456	. . . . 5 |- ( z = x -> ( ( [ z / x ] x e. A /\ C e. [_ z / x ]_ B ) <-> ( x e. A /\ C e. B ) ) )
40	34, 39	ceqsexv 2638	. . . 4 |- ( E. z ( z = x /\ ( [ z / x ] x e. A /\ C e. [_ z / x ]_ B ) ) <-> ( x e. A /\ C e. B ) )
41	33, 40	bitri 182	. . 3 |- ( E. z ( [ z / x ] x e. A /\ <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) <-> ( x e. A /\ C e. B ) )
42	24, 41	syl6bb 194	. 2 |- ( <. x , C >. e. _V -> ( <. x , C >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ C e. B ) ) )
43	1, 2, 42	pm5.21nii 652	1 |- ( <. x , C >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ C e. B ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   [wsb 1685   E.wrex 2349   _Vcvv 2601   [_csb 2908   {csn 3398   <.cop 3401   U_ciun 3678   X. cxp 4361

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-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-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

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-iun 3680  df-opab 3840  df-xp 4369

This theorem is referenced by:  eliunxp  4493  opeliunxp2  4494