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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 5763 | . 2 ⊢ dom 𝐹 = ran ◡𝐹 | |
2 | dfrn4 6058 | . 2 ⊢ ran ◡𝐹 = (◡𝐹 “ V) | |
3 | dmmpt.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | mptpreima 6091 | . 2 ⊢ (◡𝐹 “ V) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
5 | 1, 2, 4 | 3eqtri 2848 | 1 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 ↦ cmpt 5145 ◡ccnv 5553 dom cdm 5554 ran crn 5555 “ cima 5557 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-mpt 5146 df-xp 5560 df-rel 5561 df-cnv 5562 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 |
This theorem is referenced by: dmmptss 6094 dmmptg 6095 dmmptd 6492 fvmpti 6766 fvmptss 6779 fvmptss2 6792 mptexgf 6984 tz9.12lem3 9217 cardf2 9371 pmtrsn 18646 00lsp 19752 rgrx0ndm 27374 abrexexd 30268 funcnvmpt 30411 mptctf 30452 issibf 31591 rdgprc0 33038 imageval 33391 dmmptdf 41486 dmmptssf 41500 dmmptdf2 41501 dvcosre 42194 itgsinexplem1 42237 stirlinglem14 42371 fvmptrabdm 43491 |
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