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Mirrors > Home > ILE Home > Th. List > fnmpt | GIF version |
Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) |
Ref | Expression |
---|---|
mptfng.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
fnmpt | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2692 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
2 | 1 | ralimi 2493 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) |
3 | mptfng.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | mptfng 5243 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴) |
5 | 2, 4 | sylib 121 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∀wral 2414 Vcvv 2681 ↦ cmpt 3984 Fn wfn 5113 |
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-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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-fun 5120 df-fn 5121 |
This theorem is referenced by: mpt0 5245 ralrnmpt 5555 rexrnmpt 5556 fmpt 5563 fmpt2d 5575 f1ocnvd 5965 offval2 5990 ofrfval2 5991 caofinvl 5997 f1od2 6125 frectfr 6290 omfnex 6338 oeiv 6345 mptelixpg 6621 fifo 6861 efcvgfsum 11362 neif 12299 tgrest 12327 dvrecap 12835 nninfalllemn 13191 |
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