Theorem fodmrnu 4626

Description: An onto function has unique domain and range.

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		fndmu 4514	. . 3 |- ((F Fn A /\ F Fn C) -> A = C)
2		fofn 4619	. . 3 |- (F:A-onto->B -> F Fn A)
3		fofn 4619	. . 3 |- (F:C-onto->D -> F Fn C)
4	1, 2, 3	syl2an 503	. 2 |- ((F:A-onto->B /\ F:C-onto->D) -> A = C)
5		forn 4620	. . 3 |- (F:A-onto->B -> ran F = B)
6		forn 4620	. . 3 |- (F:C-onto->D -> ran F = D)
7	5, 6	sylan9req 1950	. 2 |- ((F:A-onto->B /\ F:C-onto->D) -> B = D)
8	4, 7	jca 310	1 |- ((F:A-onto->B /\ F:C-onto->D) -> (A = C /\ B = D))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298  ran crn 3987   Fn wfn 3993  -onto->wfo 3996

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-in 2603  df-ss 2605  df-fn 4009  df-f 4010  df-fo 4012

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