Theorem xrnss3v 34135

Description: A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 31985 with different symbols, cf. https://github.com/metamath/set.mm/issues/2469 . (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnss3v	|- ( A |X. B ) C_ ( _V X. ( _V X. _V ) )

Proof of Theorem xrnss3v

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-xrn 34134	. 2 |- ( A |X. B ) = ( ( `' ( 1st |` ( _V X. _V ) ) o. A ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. B ) )
2		inss1 3833	. . 3 |- ( ( `' ( 1st |` ( _V X. _V ) ) o. A ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. B ) ) C_ ( `' ( 1st |` ( _V X. _V ) ) o. A )
3		relco 5633	. . . 4 |- Rel ( `' ( 1st |` ( _V X. _V ) ) o. A )
4		vex 3203	. . . . . . . . 9 |- z e. _V
5		vex 3203	. . . . . . . . 9 |- y e. _V
6	4, 5	brcnv 5305	. . . . . . . 8 |- ( z `' ( 1st |` ( _V X. _V ) ) y <-> y ( 1st |` ( _V X. _V ) ) z )
7	4	brres 5402	. . . . . . . . 9 |- ( y ( 1st |` ( _V X. _V ) ) z <-> ( y 1st z /\ y e. ( _V X. _V ) ) )
8	7	simprbi 480	. . . . . . . 8 |- ( y ( 1st |` ( _V X. _V ) ) z -> y e. ( _V X. _V ) )
9	6, 8	sylbi 207	. . . . . . 7 |- ( z `' ( 1st |` ( _V X. _V ) ) y -> y e. ( _V X. _V ) )
10	9	adantl 482	. . . . . 6 |- ( ( x A z /\ z `' ( 1st |` ( _V X. _V ) ) y ) -> y e. ( _V X. _V ) )
11	10	exlimiv 1858	. . . . 5 |- ( E. z ( x A z /\ z `' ( 1st |` ( _V X. _V ) ) y ) -> y e. ( _V X. _V ) )
12		vex 3203	. . . . . 6 |- x e. _V
13	12, 5	opelco 5293	. . . . 5 |- ( <. x , y >. e. ( `' ( 1st |` ( _V X. _V ) ) o. A ) <-> E. z ( x A z /\ z `' ( 1st |` ( _V X. _V ) ) y ) )
14		opelxp 5146	. . . . . 6 |- ( <. x , y >. e. ( _V X. ( _V X. _V ) ) <-> ( x e. _V /\ y e. ( _V X. _V ) ) )
15	12, 14	mpbiran 953	. . . . 5 |- ( <. x , y >. e. ( _V X. ( _V X. _V ) ) <-> y e. ( _V X. _V ) )
16	11, 13, 15	3imtr4i 281	. . . 4 |- ( <. x , y >. e. ( `' ( 1st |` ( _V X. _V ) ) o. A ) -> <. x , y >. e. ( _V X. ( _V X. _V ) ) )
17	3, 16	relssi 5211	. . 3 |- ( `' ( 1st |` ( _V X. _V ) ) o. A ) C_ ( _V X. ( _V X. _V ) )
18	2, 17	sstri 3612	. 2 |- ( ( `' ( 1st |` ( _V X. _V ) ) o. A ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. B ) ) C_ ( _V X. ( _V X. _V ) )
19	1, 18	eqsstri 3635	1 |- ( A |X. B ) C_ ( _V X. ( _V X. _V ) )

Syntax hints:   /\ wa 384   E.wex 1704   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   <.cop 4183   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   1stc1st 7166   2ndc2nd 7167   |X. cxrn 33982

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-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-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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-res 5126  df-xrn 34134

This theorem is referenced by:  xrnrel  34136