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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfwlim | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
Ref | Expression |
---|---|
nfwlim.1 | ⊢ Ⅎ𝑥𝑅 |
nfwlim.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfwlim | ⊢ Ⅎ𝑥WLim(𝑅, 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-wlim 33213 | . 2 ⊢ WLim(𝑅, 𝐴) = {𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))} | |
2 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
3 | nfwlim.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
4 | nfwlim.1 | . . . . . 6 ⊢ Ⅎ𝑥𝑅 | |
5 | 3, 3, 4 | nfinf 8930 | . . . . 5 ⊢ Ⅎ𝑥inf(𝐴, 𝐴, 𝑅) |
6 | 2, 5 | nfne 3087 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅) |
7 | 4, 3, 2 | nfpred 6121 | . . . . . 6 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑦) |
8 | 7, 3, 4 | nfsup 8899 | . . . . 5 ⊢ Ⅎ𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅) |
9 | 8 | nfeq2 2972 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅) |
10 | 6, 9 | nfan 1900 | . . 3 ⊢ Ⅎ𝑥(𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)) |
11 | 10, 3 | nfrabw 3338 | . 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))} |
12 | 1, 11 | nfcxfr 2953 | 1 ⊢ Ⅎ𝑥WLim(𝑅, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 Ⅎwnfc 2936 ≠ wne 2987 {crab 3110 Predcpred 6115 supcsup 8888 infcinf 8889 WLimcwlim 33211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-sup 8890 df-inf 8891 df-wlim 33213 |
This theorem is referenced by: (None) |
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