Theorem opeliunxp 5170

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 4522	. . 3 |- U_ x e. A ( { x } X. B ) = { y | E. x e. A y e. ( { x } X. B ) }
2	1	eleq2i 2693	. 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 4932	. . 3 |- <. x , C >. e. _V
4		df-rex 2918	. . . . 5 |- ( E. x e. A y e. ( { x } X. B ) <-> E. x ( x e. A /\ y e. ( { x } X. B ) ) )
5		nfv 1843	. . . . . 6 |- F/ z ( x e. A /\ y e. ( { x } X. B ) )
6		nfs1v 2437	. . . . . . 7 |- F/ x [ z / x ] x e. A
7		nfcv 2764	. . . . . . . . 9 |- F/_ x { z }
8		nfcsb1v 3549	. . . . . . . . 9 |- F/_ x [_ z / x ]_ B
9	7, 8	nfxp 5142	. . . . . . . 8 |- F/_ x ( { z } X. [_ z / x ]_ B )
10	9	nfcri 2758	. . . . . . 7 |- F/ x y e. ( { z } X. [_ z / x ]_ B )
11	6, 10	nfan 1828	. . . . . 6 |- F/ x ( [ z / x ] x e. A /\ y e. ( { z } X. [_ z / x ]_ B ) )
12		sbequ12 2111	. . . . . . 7 |- ( x = z -> ( x e. A <-> [ z / x ] x e. A ) )
13		sneq 4187	. . . . . . . . 9 |- ( x = z -> { x } = { z } )
14		csbeq1a 3542	. . . . . . . . 9 |- ( x = z -> B = [_ z / x ]_ B )
15	13, 14	xpeq12d 5140	. . . . . . . 8 |- ( x = z -> ( { x } X. B ) = ( { z } X. [_ z / x ]_ B ) )
16	15	eleq2d 2687	. . . . . . 7 |- ( x = z -> ( y e. ( { x } X. B ) <-> y e. ( { z } X. [_ z / x ]_ B ) ) )
17	12, 16	anbi12d 747	. . . . . 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 2272	. . . . 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 264	. . . 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 2689	. . . . . 6 |- ( y = <. x , C >. -> ( y e. ( { z } X. [_ z / x ]_ B ) <-> <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) )
21	20	anbi2d 740	. . . . 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 1850	. . . 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 272	. . 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 3350	. 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 5146	. . . . . 6 |- ( <. x , C >. e. ( { z } X. [_ z / x ]_ B ) <-> ( x e. { z } /\ C e. [_ z / x ]_ B ) )
26	25	anbi2i 730	. . . . 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 838	. . . . 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		velsn 4193	. . . . . . 7 |- ( x e. { z } <-> x = z )
29		equcom 1945	. . . . . . 7 |- ( x = z <-> z = x )
30	28, 29	bitri 264	. . . . . 6 |- ( x e. { z } <-> z = x )
31	30	anbi1i 731	. . . . 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 286	. . . 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 1774	. . 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		sbequ12r 2112	. . . . 5 |- ( z = x -> ( [ z / x ] x e. A <-> x e. A ) )
35	14	equcoms 1947	. . . . . . 7 |- ( z = x -> B = [_ z / x ]_ B )
36	35	eqcomd 2628	. . . . . 6 |- ( z = x -> [_ z / x ]_ B = B )
37	36	eleq2d 2687	. . . . 5 |- ( z = x -> ( C e. [_ z / x ]_ B <-> C e. B ) )
38	34, 37	anbi12d 747	. . . 4 |- ( z = x -> ( ( [ z / x ] x e. A /\ C e. [_ z / x ]_ B ) <-> ( x e. A /\ C e. B ) ) )
39	38	equsexvw 1932	. . 3 |- ( E. z ( z = x /\ ( [ z / x ] x e. A /\ C e. [_ z / x ]_ B ) ) <-> ( x e. A /\ C e. B ) )
40	33, 39	bitri 264	. 2 |- ( E. z ( [ z / x ] x e. A /\ <. x , C >. e. ( { z } X. [_ z / x ]_ B ) ) <-> ( x e. A /\ C e. B ) )
41	2, 24, 40	3bitri 286	1 |- ( <. x , C >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ C e. B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   [wsb 1880   e. wcel 1990   {cab 2608   E.wrex 2913   [_csb 3533   {csn 4177   <.cop 4183   U_ciun 4520   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-opab 4713  df-xp 5120

This theorem is referenced by:  eliunxp  5259  opeliunxp2  5260  opeliunxp2f  7336  gsum2d2lem  18372  gsum2d2  18373  gsumcom2  18374  dprdval  18402  ptbasfi  21384  cnextfun  21868  cnextfvval  21869  cnextf  21870  dvbsss  23666  iunsnima  29428