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Mirrors > Home > NFE Home > Th. List > fodmrnu | GIF version |
Description: An onto function has unique domain and range. (Contributed by set.mm contributors, 5-Nov-2006.) |
Ref | Expression |
---|---|
fodmrnu | ⊢ ((F:A–onto→B ∧ F:C–onto→D) → (A = C ∧ B = D)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofn 5271 | . . 3 ⊢ (F:A–onto→B → F Fn A) | |
2 | fofn 5271 | . . 3 ⊢ (F:C–onto→D → F Fn C) | |
3 | fndmu 5184 | . . 3 ⊢ ((F Fn A ∧ F Fn C) → A = C) | |
4 | 1, 2, 3 | syl2an 463 | . 2 ⊢ ((F:A–onto→B ∧ F:C–onto→D) → A = C) |
5 | forn 5272 | . . 3 ⊢ (F:A–onto→B → ran F = B) | |
6 | forn 5272 | . . 3 ⊢ (F:C–onto→D → ran F = D) | |
7 | 5, 6 | sylan9req 2406 | . 2 ⊢ ((F:A–onto→B ∧ F:C–onto→D) → B = D) |
8 | 4, 7 | jca 518 | 1 ⊢ ((F:A–onto→B ∧ F:C–onto→D) → (A = C ∧ B = D)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ran crn 4773 Fn wfn 4776 –onto→wfo 4779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-fn 4790 df-f 4791 df-fo 4793 |
This theorem is referenced by: (None) |
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