Theorem fodmrnu 6123

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 6117	. . 3 |- ( F : A -onto-> B -> F Fn A )
2		fofn 6117	. . 3 |- ( F : C -onto-> D -> F Fn C )
3		fndmu 5992	. . 3 |- ( ( F Fn A /\ F Fn C ) -> A = C )
4	1, 2, 3	syl2an 494	. 2 |- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> A = C )
5		forn 6118	. . 3 |- ( F : A -onto-> B -> ran F = B )
6		forn 6118	. . 3 |- ( F : C -onto-> D -> ran F = D )
7	5, 6	sylan9req 2677	. 2 |- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> B = D )
8	4, 7	jca 554	1 |- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> ( A = C /\ B = D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   ran crn 5115   Fn wfn 5883   -onto->wfo 5886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-fn 5891  df-f 5892  df-fo 5894

This theorem is referenced by: (None)