Theorem fodmrnu 6802

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 6796	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
2		fofn 6796	. . 3 ⊢ (𝐹:𝐶–onto→𝐷 → 𝐹 Fn 𝐶)
3		fndmu 6644	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐶) → 𝐴 = 𝐶)
4	1, 2, 3	syl2an 608	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐴 = 𝐶)
5		forn 6797	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6		forn 6797	. . 3 ⊢ (𝐹:𝐶–onto→𝐷 → ran 𝐹 = 𝐷)
7	5, 6	sylan9req 2817	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐵 = 𝐷)
8	4, 7	jca 521	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ran crn 5652   Fn wfn 6532  –onto→wfo 6535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2753  df-ss 3916  df-fn 6540  df-f 6541  df-fo 6543

This theorem is used by: (None)