Theorem opeliunxp 4904

Description: Membership in a union of Cartesian 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		df-iun 4293	. . 3 |- U_ x e. A ( { x } X. B ) = { y | E. x e. A y e. ( { x } X. B ) }
2	1	eleq2i 2531	. 2 |- ( <. x , C >. e. U_ x e. A ( { x } X. B ) <-> <. x , C >. e. { y | E. x e. A y e. ( { x } X. B ) } )
3		opex 4677	. . 3 |- <. x , C >. e. _V
4		df-rex 2754	. . . . 5 |- ( E. x e. A y e. ( { x } X. B ) <-> E. x ( x e. A /\ y e. ( { x } X. B ) ) )
5		nfv 1771	. . . . . 6 |- F/ z ( x e. A /\ y e. ( { x } X. B ) )
6		nfs1v 2276	. . . . . . 7 |- F/ x [ z / x ] x e. A
7		nfcv 2602	. . . . . . . . 9 |- F/_ x { z }
8		nfcsb1v 3390	. . . . . . . . 9 |- F/_ x [_ z / x ]_ B
9	7, 8	nfxp 4879	. . . . . . . 8 |- F/_ x ( { z } X. [_ z / x ]_ B )
10	9	nfcri 2596	. . . . . . 7 |- F/ x y e. ( { z } X. [_ z / x ]_ B )
11	6, 10	nfan 2021	. . . . . 6 |- F/ x ( [ z / x ] x e. A /\ y e. ( { z } X. [_ z / x ]_ B ) )
12		sbequ12 2093	. . . . . . 7 |- ( x = z -> ( x e. A <-> [ z / x ] x e. A ) )
13		sneq 3989	. . . . . . . . 9 |- ( x = z -> { x } = { z } )
14		csbeq1a 3383	. . . . . . . . 9 |- ( x = z -> B = [_ z / x ]_ B )
15	13, 14	xpeq12d 4877	. . . . . . . 8 |- ( x = z -> ( { x } X. B ) = ( { z } X. [_ z / x ]_ B ) )
16	15	eleq2d 2524	. . . . . . 7 |- ( x = z -> ( y e. ( { x } X. B ) <-> y e. ( { z } X. [_ z / x ]_ B ) ) )
17	12, 16	anbi12d 722	. . . . . 6 |- ( x = z -> ( ( x e. A /\ y e. ( { x } X. B ) ) <-> ( [ z / x ] x e. A /\ y e. ( { z } X. [_ z / x ]_ B ) ) ) )
18	5, 11, 17	cbvex 2125	. . . . 5 |- ( E. x ( x e. A /\ y e. ( { x } X. B ) ) <-> E. z ( [ z / x ] x e. A /\ y e. ( { z } X. [_ z / x ]_ B ) ) )
19	4, 18	bitri 257	. . . 4 |- ( E. x e. A y e. ( { x } X. B ) <-> E. z ( [ z / x ] x e. A /\ y e. ( { z } X. [_ z / x ]_ B ) ) )
20		eleq1 2527	. . . . . 6 |- ( y = <. x , C >. -> ( y e. ( { z } X. [_ z / x ]_ B ) <-> <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) )
21	20	anbi2d 715	. . . . 5 |- ( 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 ) ) ) )
22	21	exbidv 1778	. . . 4 |- ( 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 ) ) ) )
23	19, 22	syl5bb 265	. . 3 |- ( 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 ) ) ) )
24	3, 23	elab 3196	. 2 |- ( <. x , C >. e. { y | E. x e. A y e. ( { x } X. B ) } <-> E. z ( [ z / x ] x e. A /\ <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) )
25		opelxp 4882	. . . . . 6 |- ( <. x , C >. e. ( { z } X. [_ z / x ]_ B ) <-> ( x e. { z } /\ C e. [_ z / x ]_ B ) )
26	25	anbi2i 705	. . . . 5 |- ( ( [ 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 811	. . . . 5 |- ( ( [ 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		elsn 3993	. . . . . . 7 |- ( x e. { z } <-> x = z )
29		equcom 1872	. . . . . . 7 |- ( x = z <-> z = x )
30	28, 29	bitri 257	. . . . . 6 |- ( x e. { z } <-> z = x )
31	30	anbi1i 706	. . . . 5 |- ( ( 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 279	. . . 4 |- ( ( [ 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 1728	. . 3 |- ( 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 3059	. . . 4 |- x e. _V
35		sbequ12r 2094	. . . . 5 |- ( z = x -> ( [ z / x ] x e. A <-> x e. A ) )
36	14	equcoms 1874	. . . . . . 7 |- ( z = x -> B = [_ z / x ]_ B )
37	36	eqcomd 2467	. . . . . 6 |- ( z = x -> [_ z / x ]_ B = B )
38	37	eleq2d 2524	. . . . 5 |- ( z = x -> ( C e. [_ z / x ]_ B <-> C e. B ) )
39	35, 38	anbi12d 722	. . . 4 |- ( z = x -> ( ( [ z / x ] x e. A /\ C e. [_ z / x ]_ B ) <-> ( x e. A /\ C e. B ) ) )
40	34, 39	ceqsexv 3095	. . 3 |- ( E. z ( z = x /\ ( [ z / x ] x e. A /\ C e. [_ z / x ]_ B ) ) <-> ( x e. A /\ C e. B ) )
41	33, 40	bitri 257	. 2 |- ( E. z ( [ z / x ] x e. A /\ <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) <-> ( x e. A /\ C e. B ) )
42	2, 24, 41	3bitri 279	1 |- ( <. x , C >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ C e. B ) )

Syntax hints:   <-> wb 189   /\ wa 375   = wceq 1454   E.wex 1673   [wsb 1807   e. wcel 1897   {cab 2447   E.wrex 2749   [_csb 3374   {csn 3979   <.cop 3985   U_ciun 4291   X. cxp 4850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-iun 4293  df-opab 4475  df-xp 4858

This theorem is referenced by:  eliunxp  4990  opeliunxp2  4991  opeliunxp2f  6982  gsum2d2lem  17653  gsum2d2  17654  gsumcom2  17655  dprdval  17683  ptbasfi  20644  cnextfun  21127  cnextfvval  21128  cnextf  21129  dvbsss  22905  iunsnima  28272