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Mirrors > Home > ILE Home > Th. List > fodmrnu | 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 5347 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
2 | fofn 5347 | . . 3 ⊢ (𝐹:𝐶–onto→𝐷 → 𝐹 Fn 𝐶) | |
3 | fndmu 5224 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐶) → 𝐴 = 𝐶) | |
4 | 1, 2, 3 | syl2an 287 | . 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐴 = 𝐶) |
5 | forn 5348 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
6 | forn 5348 | . . 3 ⊢ (𝐹:𝐶–onto→𝐷 → ran 𝐹 = 𝐷) | |
7 | 5, 6 | sylan9req 2193 | . 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → 𝐵 = 𝐷) |
8 | 4, 7 | jca 304 | 1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ran crn 4540 Fn wfn 5118 –onto→wfo 5121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 df-fn 5126 df-f 5127 df-fo 5129 |
This theorem is referenced by: (None) |
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