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Mirrors > Home > MPE Home > Th. List > nfse | Structured version Visualization version 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 |
---|---|
nffr.r | ⊢ Ⅎ𝑥𝑅 |
nffr.a | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfse | ⊢ Ⅎ𝑥 𝑅 Se 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-se 5628 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V) | |
2 | nffr.a | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2899 | . . . . . 6 ⊢ Ⅎ𝑥𝑎 | |
4 | nffr.r | . . . . . 6 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2899 | . . . . . 6 ⊢ Ⅎ𝑥𝑏 | |
6 | 3, 4, 5 | nfbr 5189 | . . . . 5 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
7 | 6, 2 | nfrabw 3464 | . . . 4 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} |
8 | 7 | nfel1 2915 | . . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V |
9 | 2, 8 | nfralw 3304 | . 2 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V |
10 | 1, 9 | nfxfr 1848 | 1 ⊢ Ⅎ𝑥 𝑅 Se 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnf 1778 ∈ wcel 2099 Ⅎwnfc 2879 ∀wral 3057 {crab 3428 Vcvv 3470 class class class wbr 5142 Se wse 5625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3058 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-se 5628 |
This theorem is referenced by: nfoi 9531 |
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