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Mirrors > Home > ILE Home > Th. List > nfdm | 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 4619 | . 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧} | |
2 | nfcv 2312 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
3 | nfrn.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | nfcv 2312 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
5 | 2, 3, 4 | nfbr 4033 | . . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧 |
6 | 5 | nfex 1630 | . . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧 |
7 | 6 | nfab 2317 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧} |
8 | 1, 7 | nfcxfr 2309 | 1 ⊢ Ⅎ𝑥dom 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∃wex 1485 {cab 2156 Ⅎwnfc 2299 class class class wbr 3987 dom cdm 4609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-dm 4619 |
This theorem is referenced by: nfrn 4854 dmiin 4855 nffn 5292 ellimc3apf 13344 |
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