Theorem brdom2g 6961

Description: Dominance relation. This variation of brdomg 6962 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 6962. (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 5547	. . 3 |- ( x = A -> ( f : x -1-1-> y <-> f : A -1-1-> y ) )
2	1	exbidv 1873	. 2 |- ( x = A -> ( E. f f : x -1-1-> y <-> E. f f : A -1-1-> y ) )
3		f1eq3 5548	. . 3 |- ( y = B -> ( f : A -1-1-> y <-> f : A -1-1-> B ) )
4	3	exbidv 1873	. 2 |- ( y = B -> ( E. f f : A -1-1-> y <-> E. f f : A -1-1-> B ) )
5		df-dom 6954	. 2 |- ~<_ = { <. x , y >. | E. f f : x -1-1-> y }
6	2, 4, 5	brabg 4369	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 1398   E.wex 1541   e. wcel 2202   class class class wbr 4093   -1-1->wf1 5330   ~<_ cdom 6951

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-fn 5336  df-f 5337  df-f1 5338  df-dom 6954

This theorem is referenced by:  f1dom4g  6969