Theorem fodmrnu 5418

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 5412	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
2		fofn 5412	. . 3 ⊢ (𝐹:𝐶–onto→𝐷 → 𝐹 Fn 𝐶)
3		fndmu 5289	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐶) → 𝐴 = 𝐶)
4	1, 2, 3	syl2an 287	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐴 = 𝐶)
5		forn 5413	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6		forn 5413	. . 3 ⊢ (𝐹:𝐶–onto→𝐷 → ran 𝐹 = 𝐷)
7	5, 6	sylan9req 2220	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐵 = 𝐷)
8	4, 7	jca 304	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  ran crn 4605   Fn wfn 5183  –onto→wfo 5186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129  df-fn 5191  df-f 5192  df-fo 5194

This theorem is referenced by: (None)