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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 6117 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
2 | fofn 6117 | . . 3 ⊢ (𝐹:𝐶–onto→𝐷 → 𝐹 Fn 𝐶) | |
3 | fndmu 5992 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐶) → 𝐴 = 𝐶) | |
4 | 1, 2, 3 | syl2an 494 | . 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐴 = 𝐶) |
5 | forn 6118 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
6 | forn 6118 | . . 3 ⊢ (𝐹:𝐶–onto→𝐷 → ran 𝐹 = 𝐷) | |
7 | 5, 6 | sylan9req 2677 | . 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐵 = 𝐷) |
8 | 4, 7 | jca 554 | 1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ran crn 5115 Fn wfn 5883 –onto→wfo 5886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-in 3581 df-ss 3588 df-fn 5891 df-f 5892 df-fo 5894 |
This theorem is referenced by: (None) |
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