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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 5909 | . 2 ⊢ dom 𝐹 = ran ◡𝐹 | |
2 | dfrn4 6224 | . 2 ⊢ ran ◡𝐹 = (◡𝐹 “ V) | |
3 | dmmpt.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | mptpreima 6260 | . 2 ⊢ (◡𝐹 “ V) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
5 | 1, 2, 4 | 3eqtri 2767 | 1 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ↦ cmpt 5231 ◡ccnv 5688 dom cdm 5689 ran crn 5690 “ cima 5692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: dmmptss 6263 dmmptg 6264 dmmptd 6714 fvmpti 7015 fvmptss 7028 fvmptss2 7042 mptexgf 7242 tz9.12lem3 9827 cardf2 9981 pmtrsn 19552 00lsp 20997 rgrx0ndm 29626 abrexexd 32537 funcnvmpt 32684 mptctf 32735 issibf 34315 rdgprc0 35775 imageval 35912 dmmptdff 45166 dmmptssf 45175 dmmptdf2 45176 dvcosre 45868 itgsinexplem1 45910 stirlinglem14 46043 fvmptrabdm 47243 |
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