Theorem fodmrnu 5458

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 5452	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
2		fofn 5452	. . 3 ⊢ (𝐹:𝐶–onto→𝐷 → 𝐹 Fn 𝐶)
3		fndmu 5329	. . 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 5453	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
6		forn 5453	. . 3 ⊢ (𝐹:𝐶–onto→𝐷 → ran 𝐹 = 𝐷)
7	5, 6	sylan9req 2241	. 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 1363  ran crn 4639   Fn wfn 5223  –onto→wfo 5226

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-in 3147  df-ss 3154  df-fn 5231  df-f 5232  df-fo 5234

This theorem is referenced by: (None)