Theorem xpinpreima2 29953

Description: Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)

Assertion

Ref	Expression
xpinpreima2	|- ( ( A C_ E /\ B C_ F ) -> ( A X. B ) = ( ( `' ( 1st |` ( E X. F ) ) " A ) i^i ( `' ( 2nd |` ( E X. F ) ) " B ) ) )

Proof of Theorem xpinpreima2

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpss 5226	. . . . . 6 |- ( E X. F ) C_ ( _V X. _V )
2		rabss2 3685	. . . . . 6 |- ( ( E X. F ) C_ ( _V X. _V ) -> { r e. ( E X. F ) | ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) } C_ { r e. ( _V X. _V ) | ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) } )
3	1, 2	mp1i 13	. . . . 5 |- ( ( A C_ E /\ B C_ F ) -> { r e. ( E X. F ) | ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) } C_ { r e. ( _V X. _V ) | ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) } )
4		simprl 794	. . . . . . 7 |- ( ( ( A C_ E /\ B C_ F ) /\ ( r e. ( _V X. _V ) /\ ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) ) ) -> r e. ( _V X. _V ) )
5		simpll 790	. . . . . . . . 9 |- ( ( ( A C_ E /\ B C_ F ) /\ ( r e. ( _V X. _V ) /\ ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) ) ) -> A C_ E )
6		simprrl 804	. . . . . . . . 9 |- ( ( ( A C_ E /\ B C_ F ) /\ ( r e. ( _V X. _V ) /\ ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) ) ) -> ( 1st ` r ) e. A )
7	5, 6	sseldd 3604	. . . . . . . 8 |- ( ( ( A C_ E /\ B C_ F ) /\ ( r e. ( _V X. _V ) /\ ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) ) ) -> ( 1st ` r ) e. E )
8		simplr 792	. . . . . . . . 9 |- ( ( ( A C_ E /\ B C_ F ) /\ ( r e. ( _V X. _V ) /\ ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) ) ) -> B C_ F )
9		simprrr 805	. . . . . . . . 9 |- ( ( ( A C_ E /\ B C_ F ) /\ ( r e. ( _V X. _V ) /\ ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) ) ) -> ( 2nd ` r ) e. B )
10	8, 9	sseldd 3604	. . . . . . . 8 |- ( ( ( A C_ E /\ B C_ F ) /\ ( r e. ( _V X. _V ) /\ ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) ) ) -> ( 2nd ` r ) e. F )
11	7, 10	jca 554	. . . . . . 7 |- ( ( ( A C_ E /\ B C_ F ) /\ ( r e. ( _V X. _V ) /\ ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) ) ) -> ( ( 1st ` r ) e. E /\ ( 2nd ` r ) e. F ) )
12		elxp7 7201	. . . . . . 7 |- ( r e. ( E X. F ) <-> ( r e. ( _V X. _V ) /\ ( ( 1st ` r ) e. E /\ ( 2nd ` r ) e. F ) ) )
13	4, 11, 12	sylanbrc 698	. . . . . 6 |- ( ( ( A C_ E /\ B C_ F ) /\ ( r e. ( _V X. _V ) /\ ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) ) ) -> r e. ( E X. F ) )
14	13	rabss3d 29351	. . . . 5 |- ( ( A C_ E /\ B C_ F ) -> { r e. ( _V X. _V ) | ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) } C_ { r e. ( E X. F ) | ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) } )
15	3, 14	eqssd 3620	. . . 4 |- ( ( A C_ E /\ B C_ F ) -> { r e. ( E X. F ) | ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) } = { r e. ( _V X. _V ) | ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) } )
16		xp2 7203	. . . 4 |- ( A X. B ) = { r e. ( _V X. _V ) | ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) }
17	15, 16	syl6reqr 2675	. . 3 |- ( ( A C_ E /\ B C_ F ) -> ( A X. B ) = { r e. ( E X. F ) | ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) } )
18		inrab 3899	. . 3 |- ( { r e. ( E X. F ) | ( 1st ` r ) e. A } i^i { r e. ( E X. F ) | ( 2nd ` r ) e. B } ) = { r e. ( E X. F ) | ( ( 1st ` r ) e. A /\ ( 2nd ` r ) e. B ) }
19	17, 18	syl6eqr 2674	. 2 |- ( ( A C_ E /\ B C_ F ) -> ( A X. B ) = ( { r e. ( E X. F ) | ( 1st ` r ) e. A } i^i { r e. ( E X. F ) | ( 2nd ` r ) e. B } ) )
20		f1stres 7190	. . . . 5 |- ( 1st |` ( E X. F ) ) : ( E X. F ) --> E
21		ffn 6045	. . . . 5 |- ( ( 1st |` ( E X. F ) ) : ( E X. F ) --> E -> ( 1st |` ( E X. F ) ) Fn ( E X. F ) )
22		fncnvima2 6339	. . . . 5 |- ( ( 1st |` ( E X. F ) ) Fn ( E X. F ) -> ( `' ( 1st |` ( E X. F ) ) " A ) = { r e. ( E X. F ) | ( ( 1st |` ( E X. F ) ) ` r ) e. A } )
23	20, 21, 22	mp2b 10	. . . 4 |- ( `' ( 1st |` ( E X. F ) ) " A ) = { r e. ( E X. F ) | ( ( 1st |` ( E X. F ) ) ` r ) e. A }
24		fvres 6207	. . . . . 6 |- ( r e. ( E X. F ) -> ( ( 1st |` ( E X. F ) ) ` r ) = ( 1st ` r ) )
25	24	eleq1d 2686	. . . . 5 |- ( r e. ( E X. F ) -> ( ( ( 1st |` ( E X. F ) ) ` r ) e. A <-> ( 1st ` r ) e. A ) )
26	25	rabbiia 3185	. . . 4 |- { r e. ( E X. F ) | ( ( 1st |` ( E X. F ) ) ` r ) e. A } = { r e. ( E X. F ) | ( 1st ` r ) e. A }
27	23, 26	eqtri 2644	. . 3 |- ( `' ( 1st |` ( E X. F ) ) " A ) = { r e. ( E X. F ) | ( 1st ` r ) e. A }
28		f2ndres 7191	. . . . 5 |- ( 2nd |` ( E X. F ) ) : ( E X. F ) --> F
29		ffn 6045	. . . . 5 |- ( ( 2nd |` ( E X. F ) ) : ( E X. F ) --> F -> ( 2nd |` ( E X. F ) ) Fn ( E X. F ) )
30		fncnvima2 6339	. . . . 5 |- ( ( 2nd |` ( E X. F ) ) Fn ( E X. F ) -> ( `' ( 2nd |` ( E X. F ) ) " B ) = { r e. ( E X. F ) | ( ( 2nd |` ( E X. F ) ) ` r ) e. B } )
31	28, 29, 30	mp2b 10	. . . 4 |- ( `' ( 2nd |` ( E X. F ) ) " B ) = { r e. ( E X. F ) | ( ( 2nd |` ( E X. F ) ) ` r ) e. B }
32		fvres 6207	. . . . . 6 |- ( r e. ( E X. F ) -> ( ( 2nd |` ( E X. F ) ) ` r ) = ( 2nd ` r ) )
33	32	eleq1d 2686	. . . . 5 |- ( r e. ( E X. F ) -> ( ( ( 2nd |` ( E X. F ) ) ` r ) e. B <-> ( 2nd ` r ) e. B ) )
34	33	rabbiia 3185	. . . 4 |- { r e. ( E X. F ) | ( ( 2nd |` ( E X. F ) ) ` r ) e. B } = { r e. ( E X. F ) | ( 2nd ` r ) e. B }
35	31, 34	eqtri 2644	. . 3 |- ( `' ( 2nd |` ( E X. F ) ) " B ) = { r e. ( E X. F ) | ( 2nd ` r ) e. B }
36	27, 35	ineq12i 3812	. 2 |- ( ( `' ( 1st |` ( E X. F ) ) " A ) i^i ( `' ( 2nd |` ( E X. F ) ) " B ) ) = ( { r e. ( E X. F ) | ( 1st ` r ) e. A } i^i { r e. ( E X. F ) | ( 2nd ` r ) e. B } )
37	19, 36	syl6eqr 2674	1 |- ( ( A C_ E /\ B C_ F ) -> ( A X. B ) = ( ( `' ( 1st |` ( E X. F ) ) " A ) i^i ( `' ( 2nd |` ( E X. F ) ) " B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  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-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  cnre2csqima  29957  sxbrsigalem2  30348  sxbrsiga  30352