Theorem fodmrnu 5600

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 5594	. . 3 |- ( F : A -onto-> B -> F Fn A )
2		fofn 5594	. . 3 |- ( F : C -onto-> D -> F Fn C )
3		fndmu 5461	. . 3 |- ( ( F Fn A /\ F Fn C ) -> A = C )
4	1, 2, 3	syl2an 289	. 2 |- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> A = C )
5		forn 5595	. . 3 |- ( F : A -onto-> B -> ran F = B )
6		forn 5595	. . 3 |- ( F : C -onto-> D -> ran F = D )
7	5, 6	sylan9req 2288	. 2 |- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> B = D )
8	4, 7	jca 306	1 |- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> ( A = C /\ B = D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   ran crn 4752   Fn wfn 5349   -onto->wfo 5352

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-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3219  df-ss 3226  df-fn 5357  df-f 5358  df-fo 5360

This theorem is referenced by: (None)