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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 5895 | . 2 ⊢ dom 𝐹 = ran ◡𝐹 | |
2 | dfrn4 6201 | . 2 ⊢ ran ◡𝐹 = (◡𝐹 “ V) | |
3 | dmmpt.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | mptpreima 6237 | . 2 ⊢ (◡𝐹 “ V) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
5 | 1, 2, 4 | 3eqtri 2763 | 1 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 {crab 3431 Vcvv 3473 ↦ cmpt 5231 ◡ccnv 5675 dom cdm 5676 ran crn 5677 “ cima 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 |
This theorem is referenced by: dmmptss 6240 dmmptg 6241 dmmptd 6695 fvmpti 6997 fvmptss 7010 fvmptss2 7023 mptexgf 7226 tz9.12lem3 9790 cardf2 9944 pmtrsn 19435 00lsp 20824 rgrx0ndm 29284 abrexexd 32180 funcnvmpt 32326 mptctf 32376 issibf 33797 rdgprc0 35236 imageval 35373 dmmptdff 44383 dmmptssf 44395 dmmptdf2 44396 dvcosre 45089 itgsinexplem1 45131 stirlinglem14 45264 fvmptrabdm 46462 |
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