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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 5238 | . 2 ⊢ dom 𝐹 = ran ◡𝐹 | |
2 | dfrn4 5513 | . 2 ⊢ ran ◡𝐹 = (◡𝐹 “ V) | |
3 | dmmpt.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | mptpreima 5545 | . 2 ⊢ (◡𝐹 “ V) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
5 | 1, 2, 4 | 3eqtri 2636 | 1 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ↦ cmpt 4643 ◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: dmmptss 5548 dmmptg 5549 dmmptd 5937 fvmpti 6190 fvmptss 6201 fvmptss2 6212 tz9.12lem3 8535 cardf2 8652 pmtrsn 17762 00lsp 18802 abrexexd 28731 mptexgf 28809 funcnvmptOLD 28850 funcnvmpt 28851 mptctf 28883 issibf 29722 rdgprc0 30943 imageval 31207 dmmptdf 38412 dvcosre 38799 itgsinexplem1 38845 stirlinglem14 38980 rgrx0ndm 40793 |
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