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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 6822 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
| 2 | fofn 6822 | . . 3 ⊢ (𝐹:𝐶–onto→𝐷 → 𝐹 Fn 𝐶) | |
| 3 | fndmu 6675 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐶) → 𝐴 = 𝐶) | |
| 4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐴 = 𝐶) |
| 5 | forn 6823 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
| 6 | forn 6823 | . . 3 ⊢ (𝐹:𝐶–onto→𝐷 → ran 𝐹 = 𝐷) | |
| 7 | 5, 6 | sylan9req 2798 | . 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐵 = 𝐷) |
| 8 | 4, 7 | jca 511 | 1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ran crn 5686 Fn wfn 6556 –onto→wfo 6559 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ss 3968 df-fn 6564 df-f 6565 df-fo 6567 |
| This theorem is referenced by: (None) |
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