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Mirrors > Home > ILE Home > Th. List > elmapi | Unicode version |
Description: A mapping is a function, forward direction only with superfluous antecedent removed. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
Ref | Expression |
---|---|
elmapi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapex 6617 | . . 3 | |
2 | elmapg 6609 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | 3 | ibi 175 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2128 cvv 2712 wf 5169 (class class class)co 5827 cmap 6596 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4029 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-fv 5181 df-ov 5830 df-oprab 5831 df-mpo 5832 df-map 6598 |
This theorem is referenced by: elmapfn 6619 elmapfun 6620 elmapssres 6621 mapsspm 6630 map0b 6635 mapss 6639 mapsncnv 6643 mapen 6794 mapxpen 6796 nninff 7069 ismkvnex 7101 bj-charfunr 13482 nninfalllem1 13677 nninfall 13678 nninfsellemdc 13679 nninfsellemqall 13684 nninfomnilem 13687 isomninnlem 13698 trilpo 13711 iswomninnlem 13717 iswomni0 13719 ismkvnnlem 13720 redcwlpo 13723 nconstwlpo 13733 neapmkv 13735 |
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