Theorem fodmrnu 5353

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 5347	. . 3 |- ( F : A -onto-> B -> F Fn A )
2		fofn 5347	. . 3 |- ( F : C -onto-> D -> F Fn C )
3		fndmu 5224	. . 3 |- ( ( F Fn A /\ F Fn C ) -> A = C )
4	1, 2, 3	syl2an 287	. 2 |- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> A = C )
5		forn 5348	. . 3 |- ( F : A -onto-> B -> ran F = B )
6		forn 5348	. . 3 |- ( F : C -onto-> D -> ran F = D )
7	5, 6	sylan9req 2193	. 2 |- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> B = D )
8	4, 7	jca 304	1 |- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> ( A = C /\ B = D ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   ran crn 4540   Fn wfn 5118   -onto->wfo 5121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084  df-fn 5126  df-f 5127  df-fo 5129

This theorem is referenced by: (None)