Theorem breng 6995

Description: Equinumerosity relation. This variation of bren 6996 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 6996. (Revised by BTernaryTau, 23-Sep-2024.)

Assertion

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

Distinct variable groups:   A, f   B, f

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

Proof of Theorem breng

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

Step	Hyp	Ref	Expression
1		f1oeq2 5608	. . 3 |- ( x = A -> ( f : x -1-1-onto-> y <-> f : A -1-1-onto-> y ) )
2	1	exbidv 1874	. 2 |- ( x = A -> ( E. f f : x -1-1-onto-> y <-> E. f f : A -1-1-onto-> y ) )
3		f1oeq3 5609	. . 3 |- ( y = B -> ( f : A -1-1-onto-> y <-> f : A -1-1-onto-> B ) )
4	3	exbidv 1874	. 2 |- ( y = B -> ( E. f f : A -1-1-onto-> y <-> E. f f : A -1-1-onto-> B ) )
5		df-en 6989	. 2 |- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
6	2, 4, 5	brabg 4392	1 |- ( ( A e. V /\ B e. W ) -> ( A ~~ B <-> E. f f : A -1-1-onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   class class class wbr 4114   -1-1-onto->wf1o 5356   ~~ cen 6986

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-en 6989

This theorem is referenced by:  f1oen4g  7004  en2prd  7072