Theorem fodmrnu 3675

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 3585	. . 3 |- ((F Fn A /\ F Fn C) -> A = C)
2		fof 3667	. . . 4 |- (F:A-onto->B -> F:A-->B)
3		ffn 3623	. . . 4 |- (F:A-->B -> F Fn A)
4	2, 3	syl 10	. . 3 |- (F:A-onto->B -> F Fn A)
5		fof 3667	. . . 4 |- (F:C-onto->D -> F:C-->D)
6		ffn 3623	. . . 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 454	. 2 |- ((F:A-onto->B /\ F:C-onto->D) -> A = C)
9		forn 3669	. . 3 |- (F:A-onto->B -> ran F = B)
10		forn 3669	. . 3 |- (F:C-onto->D -> ran F = D)
11	9, 10	sylan9req 1526	. 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 955  ran crn 3167   Fn wfn 3173  -->wf 3174  -onto->wfo 3176

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-in 2048  df-ss 2050  df-fn 3189  df-f 3190  df-fo 3192

Copyright terms: Public domain