Theorem fodmrnu 6798

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 6792	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
2		fofn 6792	. . 3 ⊢ (𝐹:𝐶–onto→𝐷 → 𝐹 Fn 𝐶)
3		fndmu 6645	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐶) → 𝐴 = 𝐶)
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐴 = 𝐶)
5		forn 6793	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6		forn 6793	. . 3 ⊢ (𝐹:𝐶–onto→𝐷 → ran 𝐹 = 𝐷)
7	5, 6	sylan9req 2791	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐵 = 𝐷)
8	4, 7	jca 511	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ran crn 5655   Fn wfn 6526  –onto→wfo 6529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2727  df-ss 3943  df-fn 6534  df-f 6535  df-fo 6537

This theorem is referenced by: (None)