Theorem wdom2d2 37602

Description: Deduction for weak dominance by a Cartesian product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)

Hypotheses

Ref	Expression
wdom2d2.a	|- ( ph -> A e. V )
wdom2d2.b	|- ( ph -> B e. W )
wdom2d2.c	|- ( ph -> C e. X )
wdom2d2.o	|- ( ( ph /\ x e. A ) -> E. y e. B E. z e. C x = X )

Assertion

Ref	Expression
wdom2d2	|- ( ph -> A ~<_* ( B X. C ) )

Distinct variable groups:   x, X   x, A   x, y, B   x, z, C, y   ph, x

Allowed substitution hints:   ph( y, z)   A( y, z)   B( z)   V( x, y, z)   W( x, y, z)   X( y, z)

Proof of Theorem wdom2d2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wdom2d2.a	. 2 |- ( ph -> A e. V )
2		wdom2d2.b	. . 3 |- ( ph -> B e. W )
3		wdom2d2.c	. . 3 |- ( ph -> C e. X )
4		xpexg 6960	. . 3 |- ( ( B e. W /\ C e. X ) -> ( B X. C ) e. _V )
5	2, 3, 4	syl2anc 693	. 2 |- ( ph -> ( B X. C ) e. _V )
6		wdom2d2.o	. . 3 |- ( ( ph /\ x e. A ) -> E. y e. B E. z e. C x = X )
7		nfcsb1v 3549	. . . . 5 |- F/_ y [_ ( 1st ` w ) / y ]_ [_ ( 2nd ` w ) / z ]_ X
8	7	nfeq2 2780	. . . 4 |- F/ y x = [_ ( 1st ` w ) / y ]_ [_ ( 2nd ` w ) / z ]_ X
9		nfcv 2764	. . . . . 6 |- F/_ z ( 1st ` w )
10		nfcsb1v 3549	. . . . . 6 |- F/_ z [_ ( 2nd ` w ) / z ]_ X
11	9, 10	nfcsb 3551	. . . . 5 |- F/_ z [_ ( 1st ` w ) / y ]_ [_ ( 2nd ` w ) / z ]_ X
12	11	nfeq2 2780	. . . 4 |- F/ z x = [_ ( 1st ` w ) / y ]_ [_ ( 2nd ` w ) / z ]_ X
13		nfv 1843	. . . 4 |- F/ w x = X
14		csbopeq1a 7221	. . . . 5 |- ( w = <. y , z >. -> [_ ( 1st ` w ) / y ]_ [_ ( 2nd ` w ) / z ]_ X = X )
15	14	eqeq2d 2632	. . . 4 |- ( w = <. y , z >. -> ( x = [_ ( 1st ` w ) / y ]_ [_ ( 2nd ` w ) / z ]_ X <-> x = X ) )
16	8, 12, 13, 15	rexxpf 5269	. . 3 |- ( E. w e. ( B X. C ) x = [_ ( 1st ` w ) / y ]_ [_ ( 2nd ` w ) / z ]_ X <-> E. y e. B E. z e. C x = X )
17	6, 16	sylibr 224	. 2 |- ( ( ph /\ x e. A ) -> E. w e. ( B X. C ) x = [_ ( 1st ` w ) / y ]_ [_ ( 2nd ` w ) / z ]_ X )
18	1, 5, 17	wdom2d 8485	1 |- ( ph -> A ~<_* ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   [_csb 3533   <.cop 4183   class class class wbr 4653   X. cxp 5112   ` cfv 5888   1stc1st 7166   2ndc2nd 7167   ~<_^* cwdom 8462

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-pw 4160  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-wdom 8464

This theorem is referenced by: (None)