Theorem domeng 6840

Description: Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146. (Contributed by NM, 24-Apr-2004.)

Assertion

Ref	Expression
domeng	|- ( B e. C -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem domeng

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4047	. 2 |- ( y = B -> ( A ~<_ y <-> A ~<_ B ) )
2		sseq2 3216	. . . 4 |- ( y = B -> ( x C_ y <-> x C_ B ) )
3	2	anbi2d 464	. . 3 |- ( y = B -> ( ( A ~~ x /\ x C_ y ) <-> ( A ~~ x /\ x C_ B ) ) )
4	3	exbidv 1847	. 2 |- ( y = B -> ( E. x ( A ~~ x /\ x C_ y ) <-> E. x ( A ~~ x /\ x C_ B ) ) )
5		vex 2774	. . 3 |- y e. _V
6	5	domen 6839	. 2 |- ( A ~<_ y <-> E. x ( A ~~ x /\ x C_ y ) )
7	1, 4, 6	vtoclbg 2833	1 |- ( B e. C -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   E.wex 1514   e. wcel 2175   C_ wss 3165   class class class wbr 4043   ~~ cen 6824   ~<_ cdom 6825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681  df-cnv 4682  df-dm 4684  df-rn 4685  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-en 6827  df-dom 6828

This theorem is referenced by:  mapdom1g  6943