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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 5741 | . 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 2148 Vcvv 2737 ↦ cmpt 4064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 |
This theorem is referenced by: mptrabex 5744 eufnfv 5747 abrexex 6117 ofmres 6136 difinfsn 7098 ctmlemr 7106 ctssdclemn0 7108 ctssdc 7111 enumct 7113 frec2uzrand 10404 frec2uzf1od 10405 frecfzennn 10425 uzennn 10435 0tonninf 10438 1tonninf 10439 hashinfom 10757 absval 11009 climle 11341 climcvg1nlem 11356 iserabs 11482 isumshft 11497 divcnv 11504 trireciplem 11507 expcnvap0 11509 expcnvre 11510 expcnv 11511 explecnv 11512 geolim 11518 geo2lim 11523 mertenslem2 11543 eftlub 11697 1arithlem1 12360 1arith 12364 ctiunct 12440 restfn 12691 peano4nninf 14725 peano3nninf 14726 nninfsellemeq 14733 nninfsellemeqinf 14735 dceqnconst 14777 dcapnconst 14778 |
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