Theorem opeliunxp 4666

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 2741	. 2 |- ( <. x , C >. e. U_ x e. A ( { x } X. B ) -> <. x , C >. e. _V )
2		opexg 4213	. 2 |- ( ( x e. A /\ C e. B ) -> <. x , C >. e. _V )
3		df-rex 2454	. . . . . 6 |- ( E. x e. A y e. ( { x } X. B ) <-> E. x ( x e. A /\ y e. ( { x } X. B ) ) )
4		nfv 1521	. . . . . . 7 |- F/ z ( x e. A /\ y e. ( { x } X. B ) )
5		nfs1v 1932	. . . . . . . 8 |- F/ x [ z / x ] x e. A
6		nfcv 2312	. . . . . . . . . 10 |- F/_ x { z }
7		nfcsb1v 3082	. . . . . . . . . 10 |- F/_ x [_ z / x ]_ B
8	6, 7	nfxp 4638	. . . . . . . . 9 |- F/_ x ( { z } X. [_ z / x ]_ B )
9	8	nfcri 2306	. . . . . . . 8 |- F/ x y e. ( { z } X. [_ z / x ]_ B )
10	5, 9	nfan 1558	. . . . . . 7 |- F/ x ( [ z / x ] x e. A /\ y e. ( { z } X. [_ z / x ]_ B ) )
11		sbequ12 1764	. . . . . . . 8 |- ( x = z -> ( x e. A <-> [ z / x ] x e. A ) )
12		sneq 3594	. . . . . . . . . 10 |- ( x = z -> { x } = { z } )
13		csbeq1a 3058	. . . . . . . . . 10 |- ( x = z -> B = [_ z / x ]_ B )
14	12, 13	xpeq12d 4636	. . . . . . . . 9 |- ( x = z -> ( { x } X. B ) = ( { z } X. [_ z / x ]_ B ) )
15	14	eleq2d 2240	. . . . . . . 8 |- ( x = z -> ( y e. ( { x } X. B ) <-> y e. ( { z } X. [_ z / x ]_ B ) ) )
16	11, 15	anbi12d 470	. . . . . . 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 1749	. . . . . 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 183	. . . . 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 2233	. . . . . . 7 |- ( y = <. x , C >. -> ( y e. ( { z } X. [_ z / x ]_ B ) <-> <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) )
20	19	anbi2d 461	. . . . . 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 1818	. . . . 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 191	. . . 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 3875	. . . 4 |- U_ x e. A ( { x } X. B ) = { y | E. x e. A y e. ( { x } X. B ) }
24	22, 23	elab2g 2877	. . 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 4641	. . . . . . 7 |- ( <. x , C >. e. ( { z } X. [_ z / x ]_ B ) <-> ( x e. { z } /\ C e. [_ z / x ]_ B ) )
26	25	anbi2i 454	. . . . . 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 556	. . . . . 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 3600	. . . . . . . 8 |- ( x e. { z } <-> x = z )
29		equcom 1699	. . . . . . . 8 |- ( x = z <-> z = x )
30	28, 29	bitri 183	. . . . . . 7 |- ( x e. { z } <-> z = x )
31	30	anbi1i 455	. . . . . 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 205	. . . . 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 1598	. . . 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 2733	. . . . 5 |- x e. _V
35		sbequ12r 1765	. . . . . 6 |- ( z = x -> ( [ z / x ] x e. A <-> x e. A ) )
36	13	equcoms 1701	. . . . . . . 8 |- ( z = x -> B = [_ z / x ]_ B )
37	36	eqcomd 2176	. . . . . . 7 |- ( z = x -> [_ z / x ]_ B = B )
38	37	eleq2d 2240	. . . . . 6 |- ( z = x -> ( C e. [_ z / x ]_ B <-> C e. B ) )
39	35, 38	anbi12d 470	. . . . 5 |- ( z = x -> ( ( [ z / x ] x e. A /\ C e. [_ z / x ]_ B ) <-> ( x e. A /\ C e. B ) ) )
40	34, 39	ceqsexv 2769	. . . 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 183	. . 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	bitrdi 195	. 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 699	1 |- ( <. x , C >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ C e. B ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   [wsb 1755   e. wcel 2141   E.wrex 2449   _Vcvv 2730   [_csb 3049   {csn 3583   <.cop 3586   U_ciun 3873   X. cxp 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-iun 3875  df-opab 4051  df-xp 4617

This theorem is referenced by:  eliunxp  4750  opeliunxp2  4751  opeliunxp2f  6217