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Mirrors > Home > NFE Home > Th. List > fof | GIF version |
Description: An onto mapping is a mapping. (Contributed by set.mm contributors, 3-Aug-1994.) |
Ref | Expression |
---|---|
fof | ⊢ (F:A–onto→B → F:A–→B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss 3323 | . . 3 ⊢ (ran F = B → ran F ⊆ B) | |
2 | 1 | anim2i 552 | . 2 ⊢ ((F Fn A ∧ ran F = B) → (F Fn A ∧ ran F ⊆ B)) |
3 | df-fo 4793 | . 2 ⊢ (F:A–onto→B ↔ (F Fn A ∧ ran F = B)) | |
4 | df-f 4791 | . 2 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B)) | |
5 | 2, 3, 4 | 3imtr4i 257 | 1 ⊢ (F:A–onto→B → F:A–→B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ⊆ wss 3257 ran crn 4773 Fn wfn 4776 –→wf 4777 –onto→wfo 4779 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-f 4791 df-fo 4793 |
This theorem is referenced by: fofun 5270 fofn 5271 dffo2 5273 foima 5274 resdif 5306 ffoss 5314 dffo4 5423 fconst5 5455 fconstfv 5456 mapsn 6026 xpassen 6057 |
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