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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 5692 | . 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 2128 Vcvv 2712 ↦ cmpt 4025 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 |
This theorem is referenced by: mptrabex 5695 eufnfv 5697 abrexex 6065 ofmres 6084 difinfsn 7044 ctmlemr 7052 ctssdclemn0 7054 ctssdc 7057 enumct 7059 frec2uzrand 10304 frec2uzf1od 10305 frecfzennn 10325 uzennn 10335 0tonninf 10338 1tonninf 10339 hashinfom 10652 absval 10901 climle 11231 climcvg1nlem 11246 iserabs 11372 isumshft 11387 divcnv 11394 trireciplem 11397 expcnvap0 11399 expcnvre 11400 expcnv 11401 explecnv 11402 geolim 11408 geo2lim 11413 mertenslem2 11433 eftlub 11587 ctiunct 12169 restfn 12355 peano4nninf 13578 peano3nninf 13579 nninfsellemeq 13586 nninfsellemeqinf 13588 dceqnconst 13630 dcapnconst 13631 |
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