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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 34258 | . 2 ⊢ WLim(𝑅, 𝐴) = {𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))} | |
2 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
3 | nfwlim.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
4 | nfwlim.1 | . . . . . 6 ⊢ Ⅎ𝑥𝑅 | |
5 | 3, 3, 4 | nfinf 9414 | . . . . 5 ⊢ Ⅎ𝑥inf(𝐴, 𝐴, 𝑅) |
6 | 2, 5 | nfne 3043 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅) |
7 | 4, 3, 2 | nfpred 6256 | . . . . . 6 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑦) |
8 | 7, 3, 4 | nfsup 9383 | . . . . 5 ⊢ Ⅎ𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅) |
9 | 8 | nfeq2 2922 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅) |
10 | 6, 9 | nfan 1902 | . . 3 ⊢ Ⅎ𝑥(𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)) |
11 | 10, 3 | nfrabw 3438 | . 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))} |
12 | 1, 11 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥WLim(𝑅, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 Ⅎwnfc 2885 ≠ wne 2941 {crab 3405 Predcpred 6250 supcsup 9372 infcinf 9373 WLimcwlim 34256 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-xp 5637 df-cnv 5639 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-sup 9374 df-inf 9375 df-wlim 34258 |
This theorem is referenced by: (None) |
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