Theorem domeng 6901

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 4087	. 2 |- ( y = B -> ( A ~<_ y <-> A ~<_ B ) )
2		sseq2 3248	. . . 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 1871	. 2 |- ( y = B -> ( E. x ( A ~~ x /\ x C_ y ) <-> E. x ( A ~~ x /\ x C_ B ) ) )
5		vex 2802	. . 3 |- y e. _V
6	5	domen 6900	. 2 |- ( A ~<_ y <-> E. x ( A ~~ x /\ x C_ y ) )
7	1, 4, 6	vtoclbg 2862	1 |- ( B e. C -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   C_ wss 3197   class class class wbr 4083   ~~ cen 6885   ~<_ cdom 6886

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-en 6888  df-dom 6889

This theorem is referenced by:  mapdom1g  7008