Theorem fodmrnu 3686

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 3595	. . 3 |- ((F Fn A /\ F Fn C) -> A = C)
2		fof 3678	. . . 4 |- (F:A-onto->B -> F:A-->B)
3		ffn 3633	. . . 4 |- (F:A-->B -> F Fn A)
4	2, 3	syl 10	. . 3 |- (F:A-onto->B -> F Fn A)
5		fof 3678	. . . 4 |- (F:C-onto->D -> F:C-->D)
6		ffn 3633	. . . 4 |- (F:C-->D -> F Fn C)
7	5, 6	syl 10	. . 3 |- (F:C-onto->D -> F Fn C)
8	1, 4, 7	syl2an 456	. 2 |- ((F:A-onto->B /\ F:C-onto->D) -> A = C)
9		forn 3680	. . 3 |- (F:A-onto->B -> ran F = B)
10		forn 3680	. . 3 |- (F:C-onto->D -> ran F = D)
11	9, 10	sylan9req 1531	. 2 |- ((F:A-onto->B /\ F:C-onto->D) -> B = D)
12	8, 11	jca 288	1 |- ((F:A-onto->B /\ F:C-onto->D) -> (A = C /\ B = D))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958  ran crn 3177   Fn wfn 3183  -->wf 3184  -onto->wfo 3186

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056  df-fn 3199  df-f 3200  df-fo 3202

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