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Mirrors > Home > ILE Home > Th. List > elmapi | GIF 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 | ⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → 𝐴:𝐶⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapex 6563 | . . 3 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) | |
2 | elmapg 6555 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ∈ (𝐵 ↑𝑚 𝐶) ↔ 𝐴:𝐶⟶𝐵)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → (𝐴 ∈ (𝐵 ↑𝑚 𝐶) ↔ 𝐴:𝐶⟶𝐵)) |
4 | 3 | ibi 175 | 1 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → 𝐴:𝐶⟶𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 Vcvv 2686 ⟶wf 5119 (class class class)co 5774 ↑𝑚 cmap 6542 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-map 6544 |
This theorem is referenced by: elmapfn 6565 elmapfun 6566 elmapssres 6567 mapsspm 6576 map0b 6581 mapss 6585 mapsncnv 6589 mapen 6740 mapxpen 6742 ismkvnex 7029 nninff 13198 nninfalllem1 13203 nninfall 13204 nninfsellemdc 13206 nninfsellemqall 13211 nninfomnilem 13214 isomninnlem 13225 trilpo 13236 |
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