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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 4160 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V) | |
2 | nfse.a | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2228 | . . . . . 6 ⊢ Ⅎ𝑥𝑎 | |
4 | nfse.r | . . . . . 6 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2228 | . . . . . 6 ⊢ Ⅎ𝑥𝑏 | |
6 | 3, 4, 5 | nfbr 3889 | . . . . 5 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
7 | 6, 2 | nfrabxy 2547 | . . . 4 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} |
8 | 7 | nfel1 2239 | . . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V |
9 | 2, 8 | nfralxy 2414 | . 2 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V |
10 | 1, 9 | nfxfr 1408 | 1 ⊢ Ⅎ𝑥 𝑅 Se 𝐴 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnf 1394 ∈ wcel 1438 Ⅎwnfc 2215 ∀wral 2359 {crab 2363 Vcvv 2619 class class class wbr 3845 Se wse 4156 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rab 2368 df-v 2621 df-un 3003 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-se 4160 |
This theorem is referenced by: (None) |
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