![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 6571 | . . 3 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) | |
2 | elmapg 6563 | . . 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 1481 Vcvv 2689 ⟶wf 5127 (class class class)co 5782 ↑𝑚 cmap 6550 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-map 6552 |
This theorem is referenced by: elmapfn 6573 elmapfun 6574 elmapssres 6575 mapsspm 6584 map0b 6589 mapss 6593 mapsncnv 6597 mapen 6748 mapxpen 6750 ismkvnex 7037 nninff 13373 nninfalllem1 13378 nninfall 13379 nninfsellemdc 13381 nninfsellemqall 13386 nninfomnilem 13389 isomninnlem 13400 trilpo 13411 iswomninnlem 13417 ismkvnnlem 13419 redcwlpo 13422 neapmkv 13425 |
Copyright terms: Public domain | W3C validator |