Theorem brdom2g 6894

Description: Dominance relation. This variation of brdomg 6895 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 6895. (Revised by BTernaryTau, 29-Nov-2024.)

Assertion

Ref	Expression
brdom2g	|- ( ( A e. V /\ B e. W ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )

Distinct variable groups:   A, f   B, f

Allowed substitution hints:   V( f)   W( f)

Proof of Theorem brdom2g

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1eq2 5526	. . 3 |- ( x = A -> ( f : x -1-1-> y <-> f : A -1-1-> y ) )
2	1	exbidv 1871	. 2 |- ( x = A -> ( E. f f : x -1-1-> y <-> E. f f : A -1-1-> y ) )
3		f1eq3 5527	. . 3 |- ( y = B -> ( f : A -1-1-> y <-> f : A -1-1-> B ) )
4	3	exbidv 1871	. 2 |- ( y = B -> ( E. f f : A -1-1-> y <-> E. f f : A -1-1-> B ) )
5		df-dom 6887	. 2 |- ~<_ = { <. x , y >. | E. f f : x -1-1-> y }
6	2, 4, 5	brabg 4356	1 |- ( ( A e. V /\ B e. W ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   class class class wbr 4082   -1-1->wf1 5314   ~<_ cdom 6884

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-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

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-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-fn 5320  df-f 5321  df-f1 5322  df-dom 6887

This theorem is referenced by:  f1dom4g  6902