Theorem breng 6915

Description: Equinumerosity relation. This variation of bren 6916 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 6916. (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 5572	. . 3 |- ( x = A -> ( f : x -1-1-onto-> y <-> f : A -1-1-onto-> y ) )
2	1	exbidv 1873	. 2 |- ( x = A -> ( E. f f : x -1-1-onto-> y <-> E. f f : A -1-1-onto-> y ) )
3		f1oeq3 5573	. . 3 |- ( y = B -> ( f : A -1-1-onto-> y <-> f : A -1-1-onto-> B ) )
4	3	exbidv 1873	. 2 |- ( y = B -> ( E. f f : A -1-1-onto-> y <-> E. f f : A -1-1-onto-> B ) )
5		df-en 6909	. 2 |- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
6	2, 4, 5	brabg 4363	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 1397   E.wex 1540   e. wcel 2202   class class class wbr 4088   -1-1-onto->wf1o 5325   ~~ cen 6906

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-en 6909

This theorem is referenced by:  f1oen4g  6924  en2prd  6991