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Mirrors > Home > MPE Home > Th. List > dmmpt | Structured version Visualization version GIF version |
Description: The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.) |
Ref | Expression |
---|---|
dmmpt.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
dmmpt | ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdm4 5552 | . 2 ⊢ dom 𝐹 = ran ◡𝐹 | |
2 | dfrn4 5840 | . 2 ⊢ ran ◡𝐹 = (◡𝐹 “ V) | |
3 | dmmpt.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | mptpreima 5873 | . 2 ⊢ (◡𝐹 “ V) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
5 | 1, 2, 4 | 3eqtri 2853 | 1 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 {crab 3121 Vcvv 3414 ↦ cmpt 4954 ◡ccnv 5345 dom cdm 5346 ran crn 5347 “ cima 5349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-mpt 4955 df-xp 5352 df-rel 5353 df-cnv 5354 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 |
This theorem is referenced by: dmmptss 5876 dmmptg 5877 dmmptd 6261 fvmpti 6532 fvmptss 6544 fvmptss2 6557 mptexgf 6746 tz9.12lem3 8936 cardf2 9089 pmtrsn 18297 00lsp 19347 rgrx0ndm 26898 abrexexd 29891 funcnvmpt 30012 mptctf 30039 issibf 30936 rdgprc0 32232 imageval 32571 dmmptdf 40218 dmmptssf 40235 dmmptdf2 40236 dvcosre 40915 itgsinexplem1 40958 stirlinglem14 41092 fvmptrabdm 42190 |
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