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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 6808 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
2 | fofn 6808 | . . 3 ⊢ (𝐹:𝐶–onto→𝐷 → 𝐹 Fn 𝐶) | |
3 | fndmu 6657 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐶) → 𝐴 = 𝐶) | |
4 | 1, 2, 3 | syl2an 597 | . 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐴 = 𝐶) |
5 | forn 6809 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
6 | forn 6809 | . . 3 ⊢ (𝐹:𝐶–onto→𝐷 → ran 𝐹 = 𝐷) | |
7 | 5, 6 | sylan9req 2794 | . 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐵 = 𝐷) |
8 | 4, 7 | jca 513 | 1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ran crn 5678 Fn wfn 6539 –onto→wfo 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3956 df-ss 3966 df-fn 6547 df-f 6548 df-fo 6550 |
This theorem is referenced by: (None) |
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