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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 5495 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V) | |
2 | nffr.a | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2897 | . . . . . 6 ⊢ Ⅎ𝑥𝑎 | |
4 | nffr.r | . . . . . 6 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2897 | . . . . . 6 ⊢ Ⅎ𝑥𝑏 | |
6 | 3, 4, 5 | nfbr 5086 | . . . . 5 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
7 | 6, 2 | nfrabw 3287 | . . . 4 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} |
8 | 7 | nfel1 2913 | . . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V |
9 | 2, 8 | nfralw 3137 | . 2 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V |
10 | 1, 9 | nfxfr 1860 | 1 ⊢ Ⅎ𝑥 𝑅 Se 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnf 1791 ∈ wcel 2112 Ⅎwnfc 2877 ∀wral 3051 {crab 3055 Vcvv 3398 class class class wbr 5039 Se wse 5492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-se 5495 |
This theorem is referenced by: nfoi 9108 |
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