Theorem domeng 6718

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 3986	. 2 |- ( y = B -> ( A ~<_ y <-> A ~<_ B ) )
2		sseq2 3166	. . . 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 1813	. 2 |- ( y = B -> ( E. x ( A ~~ x /\ x C_ y ) <-> E. x ( A ~~ x /\ x C_ B ) ) )
5		vex 2729	. . 3 |- y e. _V
6	5	domen 6717	. 2 |- ( A ~<_ y <-> E. x ( A ~~ x /\ x C_ y ) )
7	1, 4, 6	vtoclbg 2787	1 |- ( B e. C -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   C_ wss 3116   class class class wbr 3982   ~~ cen 6704   ~<_ cdom 6705

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-en 6707  df-dom 6708

This theorem is referenced by:  mapdom1g  6813