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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 5906 | . 2 ⊢ dom 𝐹 = ran ◡𝐹 | |
| 2 | dfrn4 6222 | . 2 ⊢ ran ◡𝐹 = (◡𝐹 “ V) | |
| 3 | dmmpt.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 4 | 3 | mptpreima 6258 | . 2 ⊢ (◡𝐹 “ V) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
| 5 | 1, 2, 4 | 3eqtri 2769 | 1 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ↦ cmpt 5225 ◡ccnv 5684 dom cdm 5685 ran crn 5686 “ cima 5688 |
| 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: dmmptss 6261 dmmptg 6262 dmmptd 6713 fvmpti 7015 fvmptss 7028 fvmptss2 7042 mptexgf 7242 tz9.12lem3 9829 cardf2 9983 pmtrsn 19537 00lsp 20979 rgrx0ndm 29611 abrexexd 32528 funcnvmpt 32677 mptctf 32729 issibf 34335 rdgprc0 35794 imageval 35931 dmmptdff 45228 dmmptssf 45237 dmmptdf2 45238 dvcosre 45927 itgsinexplem1 45969 stirlinglem14 46102 fvmptrabdm 47305 |
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