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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 31002 | . 2 ⊢ WLim(𝑅, 𝐴) = {𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))} | |
2 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
3 | nfwlim.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
4 | nfwlim.1 | . . . . . 6 ⊢ Ⅎ𝑥𝑅 | |
5 | 3, 3, 4 | nfinf 8271 | . . . . 5 ⊢ Ⅎ𝑥inf(𝐴, 𝐴, 𝑅) |
6 | 2, 5 | nfne 2882 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅) |
7 | 4, 3, 2 | nfpred 5602 | . . . . . 6 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑦) |
8 | 7, 3, 4 | nfsup 8240 | . . . . 5 ⊢ Ⅎ𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅) |
9 | 8 | nfeq2 2766 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅) |
10 | 6, 9 | nfan 1816 | . . 3 ⊢ Ⅎ𝑥(𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)) |
11 | 10, 3 | nfrab 3100 | . 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))} |
12 | 1, 11 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑥WLim(𝑅, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 Ⅎwnfc 2738 ≠ wne 2780 {crab 2900 Predcpred 5596 supcsup 8229 infcinf 8230 WLimcwlim 30998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-sup 8231 df-inf 8232 df-wlim 31002 |
This theorem is referenced by: (None) |
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