Theorem fodmrnu 5448

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 5442	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
2		fofn 5442	. . 3 ⊢ (𝐹:𝐶–onto→𝐷 → 𝐹 Fn 𝐶)
3		fndmu 5319	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐶) → 𝐴 = 𝐶)
4	1, 2, 3	syl2an 289	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐴 = 𝐶)
5		forn 5443	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6		forn 5443	. . 3 ⊢ (𝐹:𝐶–onto→𝐷 → ran 𝐹 = 𝐷)
7	5, 6	sylan9req 2231	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐵 = 𝐷)
8	4, 7	jca 306	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353  ran crn 4629   Fn wfn 5213  –onto→wfo 5216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144  df-fn 5221  df-f 5222  df-fo 5224

This theorem is referenced by: (None)