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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 6087 | . 2 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
3 | 2 | ssrab3 4054 | 1 ⊢ dom 𝐹 ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 ↦ cmpt 5137 dom cdm 5548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-mpt 5138 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 |
This theorem is referenced by: mptrcl 6769 fvmptss 6772 fvmptex 6774 fvmptnf 6782 elfvmptrab1w 6786 elfvmptrab1 6787 mptexg 6975 mptexw 7643 dmmpossx 7753 tposssxp 7885 mptfi 8811 cnvimamptfin 8813 cantnfres 9128 mptct 9948 arwrcl 17292 cntzrcl 18395 gsumconst 18983 psrass1lem 20085 psrass1 20113 psrass23l 20116 psrcom 20117 psrass23 20118 mpfrcl 20226 psropprmul 20334 coe1mul2 20365 lmrcl 21767 1stcrestlem 21988 ptbasfi 22117 isxms2 22985 setsmstopn 23015 tngtopn 23186 rrxmval 23935 ulmss 24912 dchrrcl 25743 gsummpt2co 30613 locfinreflem 31003 sitgclg 31499 cvmsrcl 32408 snmlval 32475 gonan0 32536 bj-fvmptunsn1 34431 eldiophb 39232 elmnc 39614 itgocn 39642 submgmrcl 43926 dmmpossx2 44313 |
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