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Mirrors > Home > MPE Home > Th. List > fodmrnu | Structured version Visualization version GIF version |
Description: An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.) |
Ref | Expression |
---|---|
fodmrnu | ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofn 6567 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
2 | fofn 6567 | . . 3 ⊢ (𝐹:𝐶–onto→𝐷 → 𝐹 Fn 𝐶) | |
3 | fndmu 6429 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐶) → 𝐴 = 𝐶) | |
4 | 1, 2, 3 | syl2an 598 | . 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐴 = 𝐶) |
5 | forn 6568 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
6 | forn 6568 | . . 3 ⊢ (𝐹:𝐶–onto→𝐷 → ran 𝐹 = 𝐷) | |
7 | 5, 6 | sylan9req 2854 | . 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐵 = 𝐷) |
8 | 4, 7 | jca 515 | 1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ran crn 5520 Fn wfn 6319 –onto→wfo 6322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-fn 6327 df-f 6328 df-fo 6330 |
This theorem is referenced by: (None) |
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