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| Mirrors > Home > MPE Home > Th. List > dmmptss | Structured version Visualization version GIF version | ||
| Description: The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.) |
| Ref | Expression |
|---|---|
| dmmpt.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
| Ref | Expression |
|---|---|
| dmmptss | ⊢ dom 𝐹 ⊆ 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmmpt.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 2 | 1 | dmmpt 6260 | . 2 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
| 3 | 2 | ssrab3 4082 | 1 ⊢ dom 𝐹 ⊆ 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ↦ cmpt 5225 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: mptrcl 7025 fvmptss 7028 fvmptex 7030 fvmptnf 7038 elfvmptrab1w 7043 elfvmptrab1 7044 mptexg 7241 mptexw 7977 dmmpossx 8091 tposssxp 8255 mptfi 9391 cnvimamptfin 9393 cantnfres 9717 mptct 10578 arwrcl 18089 submgmrcl 18708 cntzrcl 19345 gsumconst 19952 psrass1lem 21952 psrass1 21984 psrass23l 21987 psrcom 21988 psrass23 21989 mpfrcl 22109 psropprmul 22239 coe1mul2 22272 lmrcl 23239 1stcrestlem 23460 ptbasfi 23589 isxms2 24458 setsmstopn 24490 tngtopn 24671 rrxmval 25439 ulmss 26440 dchrrcl 27284 gsummpt2co 33051 locfinreflem 33839 sitgclg 34344 cvmsrcl 35269 snmlval 35336 gonan0 35397 bj-fvmptunsn1 37258 eldiophb 42768 elmnc 43148 itgocn 43176 dmmpossx2 48253 dmtposss 48776 |
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