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Mirrors > Home > MPE Home > Th. List > nfdm | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
nfrn.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfdm | ⊢ Ⅎ𝑥dom 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dm 5641 | . 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧} | |
2 | nfcv 2906 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
3 | nfrn.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | nfcv 2906 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
5 | 2, 3, 4 | nfbr 5151 | . . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧 |
6 | 5 | nfex 2319 | . . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧 |
7 | 6 | nfab 2912 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧} |
8 | 1, 7 | nfcxfr 2904 | 1 ⊢ Ⅎ𝑥dom 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1782 {cab 2715 Ⅎwnfc 2886 class class class wbr 5104 dom cdm 5631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-dm 5641 |
This theorem is referenced by: nfrn 5904 dmiin 5905 nffn 6597 nosupbnd2 26986 noinfbnd2 27001 funimass4f 31336 bnj1398 33407 bnj1491 33430 fnlimcnv 43618 fnlimfvre 43625 fnlimabslt 43630 lmbr3 43698 itgsinexplem1 43905 fourierdlem16 44074 fourierdlem21 44079 fourierdlem22 44080 fourierdlem68 44125 fourierdlem80 44137 fourierdlem103 44160 fourierdlem104 44161 issmff 44683 issmfdf 44686 smfpimltmpt 44695 smfpimltxr 44696 smfpimltxrmptf 44707 smfpreimagtf 44717 smflim 44726 smfpimgtxr 44729 smfpimgtmpt 44730 smfpimgtxrmptf 44733 smflim2 44755 smfpimcc 44757 smfsup 44763 smfsupmpt 44764 smfsupxr 44765 smfinflem 44766 smfinf 44767 smfinfmpt 44768 smflimsup 44777 smfliminf 44780 adddmmbl2 44783 muldmmbl2 44785 smfpimne2 44789 smfdivdmmbl2 44790 nfdfat 45059 |
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