Theorem brdom2g 7021

Description: Dominance relation. This variation of brdomg 7022 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 7022. (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 5589	. . 3 |- ( x = A -> ( f : x -1-1-> y <-> f : A -1-1-> y ) )
2	1	exbidv 1878	. 2 |- ( x = A -> ( E. f f : x -1-1-> y <-> E. f f : A -1-1-> y ) )
3		f1eq3 5590	. . 3 |- ( y = B -> ( f : A -1-1-> y <-> f : A -1-1-> B ) )
4	3	exbidv 1878	. 2 |- ( y = B -> ( E. f f : A -1-1-> y <-> E. f f : A -1-1-> B ) )
5		df-dom 7014	. 2 |- ~<_ = { <. x , y >. | E. f f : x -1-1-> y }
6	2, 4, 5	brabg 4406	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 1402   E.wex 1545   e. wcel 2209   class class class wbr 4125   -1-1->wf1 5369   ~<_ cdom 7011

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-fn 5375  df-f 5376  df-f1 5377  df-dom 7014

This theorem is referenced by:  f1dom4g  7029