Theorem opeliunxp 4781

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 2814	. 2 |- ( <. x , C >. e. U_ x e. A ( { x } X. B ) -> <. x , C >. e. _V )
2		opexg 4320	. 2 |- ( ( x e. A /\ C e. B ) -> <. x , C >. e. _V )
3		df-rex 2516	. . . . . 6 |- ( E. x e. A y e. ( { x } X. B ) <-> E. x ( x e. A /\ y e. ( { x } X. B ) ) )
4		nfv 1576	. . . . . . 7 |- F/ z ( x e. A /\ y e. ( { x } X. B ) )
5		nfs1v 1992	. . . . . . . 8 |- F/ x [ z / x ] x e. A
6		nfcv 2374	. . . . . . . . . 10 |- F/_ x { z }
7		nfcsb1v 3160	. . . . . . . . . 10 |- F/_ x [_ z / x ]_ B
8	6, 7	nfxp 4752	. . . . . . . . 9 |- F/_ x ( { z } X. [_ z / x ]_ B )
9	8	nfcri 2368	. . . . . . . 8 |- F/ x y e. ( { z } X. [_ z / x ]_ B )
10	5, 9	nfan 1613	. . . . . . 7 |- F/ x ( [ z / x ] x e. A /\ y e. ( { z } X. [_ z / x ]_ B ) )
11		sbequ12 1819	. . . . . . . 8 |- ( x = z -> ( x e. A <-> [ z / x ] x e. A ) )
12		sneq 3680	. . . . . . . . . 10 |- ( x = z -> { x } = { z } )
13		csbeq1a 3136	. . . . . . . . . 10 |- ( x = z -> B = [_ z / x ]_ B )
14	12, 13	xpeq12d 4750	. . . . . . . . 9 |- ( x = z -> ( { x } X. B ) = ( { z } X. [_ z / x ]_ B ) )
15	14	eleq2d 2301	. . . . . . . 8 |- ( x = z -> ( y e. ( { x } X. B ) <-> y e. ( { z } X. [_ z / x ]_ B ) ) )
16	11, 15	anbi12d 473	. . . . . . 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 1804	. . . . . 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 184	. . . . 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 2294	. . . . . . 7 |- ( y = <. x , C >. -> ( y e. ( { z } X. [_ z / x ]_ B ) <-> <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) )
20	19	anbi2d 464	. . . . . 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 1873	. . . . 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	bitrid 192	. . . 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 3972	. . . 4 |- U_ x e. A ( { x } X. B ) = { y | E. x e. A y e. ( { x } X. B ) }
24	22, 23	elab2g 2953	. . 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 4755	. . . . . . 7 |- ( <. x , C >. e. ( { z } X. [_ z / x ]_ B ) <-> ( x e. { z } /\ C e. [_ z / x ]_ B ) )
26	25	anbi2i 457	. . . . . 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 563	. . . . . 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 3686	. . . . . . . 8 |- ( x e. { z } <-> x = z )
29		equcom 1754	. . . . . . . 8 |- ( x = z <-> z = x )
30	28, 29	bitri 184	. . . . . . 7 |- ( x e. { z } <-> z = x )
31	30	anbi1i 458	. . . . . 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 206	. . . . 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 1653	. . . 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 2805	. . . . 5 |- x e. _V
35		sbequ12r 1820	. . . . . 6 |- ( z = x -> ( [ z / x ] x e. A <-> x e. A ) )
36	13	equcoms 1756	. . . . . . . 8 |- ( z = x -> B = [_ z / x ]_ B )
37	36	eqcomd 2237	. . . . . . 7 |- ( z = x -> [_ z / x ]_ B = B )
38	37	eleq2d 2301	. . . . . 6 |- ( z = x -> ( C e. [_ z / x ]_ B <-> C e. B ) )
39	35, 38	anbi12d 473	. . . . 5 |- ( z = x -> ( ( [ z / x ] x e. A /\ C e. [_ z / x ]_ B ) <-> ( x e. A /\ C e. B ) ) )
40	34, 39	ceqsexv 2842	. . . 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 184	. . 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 196	. 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 711	1 |- ( <. x , C >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ C e. B ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   [wsb 1810   e. wcel 2202   E.wrex 2511   _Vcvv 2802   [_csb 3127   {csn 3669   <.cop 3672   U_ciun 3970   X. cxp 4723

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-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-iun 3972  df-opab 4151  df-xp 4731

This theorem is referenced by:  eliunxp  4869  opeliunxp2  4870  opeliunxp2f  6403