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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 6627 | . . 3 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) | |
2 | elmapg 6619 | . . 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 2135 Vcvv 2722 ⟶wf 5179 (class class class)co 5837 ↑𝑚 cmap 6606 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-opab 4039 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-fv 5191 df-ov 5840 df-oprab 5841 df-mpo 5842 df-map 6608 |
This theorem is referenced by: elmapfn 6629 elmapfun 6630 elmapssres 6631 mapsspm 6640 map0b 6645 mapss 6649 mapsncnv 6653 mapen 6804 mapxpen 6806 nninff 7079 ismkvnex 7111 bj-charfunr 13544 nninfalllem1 13740 nninfall 13741 nninfsellemdc 13742 nninfsellemqall 13747 nninfomnilem 13750 isomninnlem 13761 trilpo 13774 iswomninnlem 13780 iswomni0 13782 ismkvnnlem 13783 redcwlpo 13786 nconstwlpo 13796 neapmkv 13798 |
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