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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 4373 | . 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧} | |
2 | nfcv 2219 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
3 | nfrn.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | nfcv 2219 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
5 | 2, 3, 4 | nfbr 3829 | . . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧 |
6 | 5 | nfex 1568 | . . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧 |
7 | 6 | nfab 2223 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧} |
8 | 1, 7 | nfcxfr 2216 | 1 ⊢ Ⅎ𝑥dom 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∃wex 1421 {cab 2067 Ⅎwnfc 2206 class class class wbr 3785 dom cdm 4363 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-dm 4373 |
This theorem is referenced by: nfrn 4597 dmiin 4598 nffn 5015 |
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