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Mirrors > Home > ILE Home > Th. List > nfse | GIF version |
Description: Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
nfse.r | ⊢ Ⅎ𝑥𝑅 |
nfse.a | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfse | ⊢ Ⅎ𝑥 𝑅 Se 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-se 4327 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V) | |
2 | nfse.a | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2317 | . . . . . 6 ⊢ Ⅎ𝑥𝑎 | |
4 | nfse.r | . . . . . 6 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2317 | . . . . . 6 ⊢ Ⅎ𝑥𝑏 | |
6 | 3, 4, 5 | nfbr 4044 | . . . . 5 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
7 | 6, 2 | nfrabxy 2655 | . . . 4 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} |
8 | 7 | nfel1 2328 | . . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V |
9 | 2, 8 | nfralxy 2513 | . 2 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V |
10 | 1, 9 | nfxfr 1472 | 1 ⊢ Ⅎ𝑥 𝑅 Se 𝐴 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnf 1458 ∈ wcel 2146 Ⅎwnfc 2304 ∀wral 2453 {crab 2457 Vcvv 2735 class class class wbr 3998 Se wse 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-ext 2157 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rab 2462 df-v 2737 df-un 3131 df-sn 3595 df-pr 3596 df-op 3598 df-br 3999 df-se 4327 |
This theorem is referenced by: (None) |
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