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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 43630 fnlimfvre 43637 fnlimabslt 43642 lmbr3 43710 itgsinexplem1 43917 fourierdlem16 44086 fourierdlem21 44091 fourierdlem22 44092 fourierdlem68 44137 fourierdlem80 44149 fourierdlem103 44172 fourierdlem104 44173 issmff 44697 issmfdf 44700 smfpimltmpt 44709 smfpimltxr 44710 smfpimltxrmptf 44721 smfpreimagtf 44731 smflim 44740 smfpimgtxr 44743 smfpimgtmpt 44744 smfpimgtxrmptf 44747 smflim2 44769 smfpimcc 44771 smfsup 44777 smfsupmpt 44778 smfsupxr 44779 smfinflem 44780 smfinf 44781 smfinfmpt 44782 smflimsup 44791 smfliminf 44794 adddmmbl2 44797 muldmmbl2 44799 smfpimne2 44803 smfdivdmmbl2 44804 fsupdm 44805 nfdfat 45077 |
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