Theorem domeng 6654

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 3941	. 2 |- ( y = B -> ( A ~<_ y <-> A ~<_ B ) )
2		sseq2 3126	. . . 4 |- ( y = B -> ( x C_ y <-> x C_ B ) )
3	2	anbi2d 460	. . 3 |- ( y = B -> ( ( A ~~ x /\ x C_ y ) <-> ( A ~~ x /\ x C_ B ) ) )
4	3	exbidv 1798	. 2 |- ( y = B -> ( E. x ( A ~~ x /\ x C_ y ) <-> E. x ( A ~~ x /\ x C_ B ) ) )
5		vex 2692	. . 3 |- y e. _V
6	5	domen 6653	. 2 |- ( A ~<_ y <-> E. x ( A ~~ x /\ x C_ y ) )
7	1, 4, 6	vtoclbg 2750	1 |- ( B e. C -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   C_ wss 3076   class class class wbr 3937   ~~ cen 6640   ~<_ cdom 6641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-en 6643  df-dom 6644

This theorem is referenced by:  mapdom1g  6749