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Mirrors > Home > ILE Home > Th. List > mptex | GIF version |
Description: If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.) |
Ref | Expression |
---|---|
mptex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
mptex | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | mptexg 5638 | . 2 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Vcvv 2681 ↦ cmpt 3984 |
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-coll 4038 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-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 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-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 |
This theorem is referenced by: eufnfv 5641 abrexex 6008 ofmres 6027 difinfsn 6978 ctmlemr 6986 ctssdclemn0 6988 ctssdc 6991 enumct 6993 frec2uzrand 10171 frec2uzf1od 10172 frecfzennn 10192 uzennn 10202 0tonninf 10205 1tonninf 10206 hashinfom 10517 absval 10766 climle 11096 climcvg1nlem 11111 iserabs 11237 isumshft 11252 divcnv 11259 trireciplem 11262 expcnvap0 11264 expcnvre 11265 expcnv 11266 explecnv 11267 geolim 11273 geo2lim 11278 mertenslem2 11298 eftlub 11385 ctiunct 11942 restfn 12113 peano4nninf 13189 peano3nninf 13190 nninfsellemeq 13199 nninfsellemeqinf 13201 |
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