Theorem fodmrnu 5506

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 5500	. . 3 |- ( F : A -onto-> B -> F Fn A )
2		fofn 5500	. . 3 |- ( F : C -onto-> D -> F Fn C )
3		fndmu 5377	. . 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 5501	. . 3 |- ( F : A -onto-> B -> ran F = B )
6		forn 5501	. . 3 |- ( F : C -onto-> D -> ran F = D )
7	5, 6	sylan9req 2259	. 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 1373   ran crn 4676   Fn wfn 5266   -onto->wfo 5269

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179  df-fn 5274  df-f 5275  df-fo 5277

This theorem is referenced by: (None)