Theorem fodmrnu 5134

Description: An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)

Assertion

Ref	Expression
fodmrnu	|- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> ( A = C /\ B = D ) )

Proof of Theorem fodmrnu

Step	Hyp	Ref	Expression
1		fofn 5128	. . 3 |- ( F : A -onto-> B -> F Fn A )
2		fofn 5128	. . 3 |- ( F : C -onto-> D -> F Fn C )
3		fndmu 5020	. . 3 |- ( ( F Fn A /\ F Fn C ) -> A = C )
4	1, 2, 3	syl2an 283	. 2 |- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> A = C )
5		forn 5129	. . 3 |- ( F : A -onto-> B -> ran F = B )
6		forn 5129	. . 3 |- ( F : C -onto-> D -> ran F = D )
7	5, 6	sylan9req 2134	. 2 |- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> B = D )
8	4, 7	jca 300	1 |- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> ( A = C /\ B = D ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   ran crn 4364   Fn wfn 4917   -onto->wfo 4920

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986  df-fn 4925  df-f 4926  df-fo 4928

This theorem is referenced by: (None)