Theorem breng 6833

Description: Equinumerosity relation. This variation of bren 6834 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 6834. (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 5510	. . 3 |- ( x = A -> ( f : x -1-1-onto-> y <-> f : A -1-1-onto-> y ) )
2	1	exbidv 1847	. 2 |- ( x = A -> ( E. f f : x -1-1-onto-> y <-> E. f f : A -1-1-onto-> y ) )
3		f1oeq3 5511	. . 3 |- ( y = B -> ( f : A -1-1-onto-> y <-> f : A -1-1-onto-> B ) )
4	3	exbidv 1847	. 2 |- ( y = B -> ( E. f f : A -1-1-onto-> y <-> E. f f : A -1-1-onto-> B ) )
5		df-en 6827	. 2 |- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
6	2, 4, 5	brabg 4314	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 1372   E.wex 1514   e. wcel 2175   class class class wbr 4043   -1-1-onto->wf1o 5269   ~~ cen 6824

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-en 6827

This theorem is referenced by:  f1oen4g  6842  en2prd  6908