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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 5695 | . 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧} | |
| 2 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
| 3 | nfrn.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
| 4 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
| 5 | 2, 3, 4 | nfbr 5190 | . . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧 |
| 6 | 5 | nfex 2324 | . . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧 |
| 7 | 6 | nfab 2911 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧} |
| 8 | 1, 7 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥dom 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ∃wex 1779 {cab 2714 Ⅎwnfc 2890 class class class wbr 5143 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-dm 5695 |
| This theorem is referenced by: nfrn 5963 dmiin 5964 nffn 6667 nosupbnd2 27761 noinfbnd2 27776 funimass4f 32647 bnj1398 35048 bnj1491 35071 fnlimcnv 45682 fnlimfvre 45689 fnlimabslt 45694 lmbr3 45762 itgsinexplem1 45969 fourierdlem16 46138 fourierdlem21 46143 fourierdlem22 46144 fourierdlem68 46189 fourierdlem80 46201 fourierdlem103 46224 fourierdlem104 46225 issmff 46749 issmfdf 46752 smfpimltmpt 46761 smfpimltxr 46762 smfpimltxrmptf 46773 smfpreimagtf 46783 smflim 46792 smfpimgtxr 46795 smfpimgtmpt 46796 smfpimgtxrmptf 46799 smflim2 46821 smfpimcc 46823 smfsup 46829 smfsupmpt 46830 smfsupxr 46831 smfinflem 46832 smfinf 46833 smflimsup 46843 smfliminf 46846 adddmmbl2 46849 muldmmbl2 46851 smfpimne2 46855 smfdivdmmbl2 46856 fsupdm 46857 finfdm 46861 nfdfat 47139 |
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