Theorem elxp8 33219

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7201. (Contributed by ML, 19-Oct-2020.)

Assertion

Ref	Expression
elxp8	|- ( A e. ( B X. C ) <-> ( ( 1st ` A ) e. B /\ A e. ( _V X. C ) ) )

Proof of Theorem elxp8

Step	Hyp	Ref	Expression
1		xp1st 7198	. . 3 |- ( A e. ( B X. C ) -> ( 1st ` A ) e. B )
2		ssv 3625	. . . . 5 |- B C_ _V
3		ssid 3624	. . . . 5 |- C C_ C
4		xpss12 5225	. . . . 5 |- ( ( B C_ _V /\ C C_ C ) -> ( B X. C ) C_ ( _V X. C ) )
5	2, 3, 4	mp2an 708	. . . 4 |- ( B X. C ) C_ ( _V X. C )
6	5	sseli 3599	. . 3 |- ( A e. ( B X. C ) -> A e. ( _V X. C ) )
7	1, 6	jca 554	. 2 |- ( A e. ( B X. C ) -> ( ( 1st ` A ) e. B /\ A e. ( _V X. C ) ) )
8		xpss 5226	. . . . 5 |- ( _V X. C ) C_ ( _V X. _V )
9	8	sseli 3599	. . . 4 |- ( A e. ( _V X. C ) -> A e. ( _V X. _V ) )
10	9	adantl 482	. . 3 |- ( ( ( 1st ` A ) e. B /\ A e. ( _V X. C ) ) -> A e. ( _V X. _V ) )
11		xp2nd 7199	. . . 4 |- ( A e. ( _V X. C ) -> ( 2nd ` A ) e. C )
12	11	anim2i 593	. . 3 |- ( ( ( 1st ` A ) e. B /\ A e. ( _V X. C ) ) -> ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) )
13		elxp7 7201	. . 3 |- ( A e. ( B X. C ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
14	10, 12, 13	sylanbrc 698	. 2 |- ( ( ( 1st ` A ) e. B /\ A e. ( _V X. C ) ) -> A e. ( B X. C ) )
15	7, 14	impbii 199	1 |- ( A e. ( B X. C ) <-> ( ( 1st ` A ) e. B /\ A e. ( _V X. C ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   C_ wss 3574   X. cxp 5112   ` 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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-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-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  finxpsuclem  33234