Theorem breng 6847

Description: Equinumerosity relation. This variation of bren 6848 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 6848. (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 5523	. . 3 |- ( x = A -> ( f : x -1-1-onto-> y <-> f : A -1-1-onto-> y ) )
2	1	exbidv 1849	. 2 |- ( x = A -> ( E. f f : x -1-1-onto-> y <-> E. f f : A -1-1-onto-> y ) )
3		f1oeq3 5524	. . 3 |- ( y = B -> ( f : A -1-1-onto-> y <-> f : A -1-1-onto-> B ) )
4	3	exbidv 1849	. 2 |- ( y = B -> ( E. f f : A -1-1-onto-> y <-> E. f f : A -1-1-onto-> B ) )
5		df-en 6841	. 2 |- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
6	2, 4, 5	brabg 4323	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 1373   E.wex 1516   e. wcel 2177   class class class wbr 4051   -1-1-onto->wf1o 5279   ~~ cen 6838

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-en 6841

This theorem is referenced by:  f1oen4g  6856  en2prd  6923