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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 5721 | . 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 2141 Vcvv 2730 ↦ cmpt 4050 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 |
This theorem is referenced by: mptrabex 5724 eufnfv 5726 abrexex 6096 ofmres 6115 difinfsn 7077 ctmlemr 7085 ctssdclemn0 7087 ctssdc 7090 enumct 7092 frec2uzrand 10361 frec2uzf1od 10362 frecfzennn 10382 uzennn 10392 0tonninf 10395 1tonninf 10396 hashinfom 10712 absval 10965 climle 11297 climcvg1nlem 11312 iserabs 11438 isumshft 11453 divcnv 11460 trireciplem 11463 expcnvap0 11465 expcnvre 11466 expcnv 11467 explecnv 11468 geolim 11474 geo2lim 11479 mertenslem2 11499 eftlub 11653 1arithlem1 12315 1arith 12319 ctiunct 12395 restfn 12583 peano4nninf 14039 peano3nninf 14040 nninfsellemeq 14047 nninfsellemeqinf 14049 dceqnconst 14091 dcapnconst 14092 |
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