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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 6251 | . 2 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
3 | 2 | ssrab3 4079 | 1 ⊢ dom 𝐹 ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3947 ↦ cmpt 5236 dom cdm 5682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5154 df-opab 5216 df-mpt 5237 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 |
This theorem is referenced by: mptrcl 7018 fvmptss 7021 fvmptex 7023 fvmptnf 7031 elfvmptrab1w 7036 elfvmptrab1 7037 mptexg 7238 mptexw 7966 dmmpossx 8080 tposssxp 8245 mptfi 9395 cnvimamptfin 9397 cantnfres 9720 mptct 10581 arwrcl 18066 submgmrcl 18688 cntzrcl 19321 gsumconst 19932 psrass1lemOLD 21938 psrass1lem 21941 psrass1 21973 psrass23l 21976 psrcom 21977 psrass23 21978 mpfrcl 22100 psropprmul 22227 coe1mul2 22260 lmrcl 23226 1stcrestlem 23447 ptbasfi 23576 isxms2 24445 setsmstopn 24477 tngtopn 24658 rrxmval 25424 ulmss 26426 dchrrcl 27269 gsummpt2co 32916 locfinreflem 33655 sitgclg 34176 cvmsrcl 35092 snmlval 35159 gonan0 35220 bj-fvmptunsn1 36964 eldiophb 42414 elmnc 42797 itgocn 42825 dmmpossx2 47715 |
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