Theorem fodmrnu 5278

Description: An onto function has unique domain and range. (Contributed by set.mm contributors, 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 5272	. . 3 ⊢ (F:A–onto→B → F Fn A)
2		fofn 5272	. . 3 ⊢ (F:C–onto→D → F Fn C)
3		fndmu 5185	. . 3 ⊢ ((F Fn A ∧ F Fn C) → A = C)
4	1, 2, 3	syl2an 463	. 2 ⊢ ((F:A–onto→B ∧ F:C–onto→D) → A = C)
5		forn 5273	. . 3 ⊢ (F:A–onto→B → ran F = B)
6		forn 5273	. . 3 ⊢ (F:C–onto→D → ran F = D)
7	5, 6	sylan9req 2406	. 2 ⊢ ((F:A–onto→B ∧ F:C–onto→D) → B = D)
8	4, 7	jca 518	1 ⊢ ((F:A–onto→B ∧ F:C–onto→D) → (A = C ∧ B = D))

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  ran crn 4774   Fn wfn 4777  –onto→wfo 4780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-fn 4791  df-f 4792  df-fo 4794

This theorem is referenced by: (None)