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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 35637 | . 2 ⊢ WLim(𝑅, 𝐴) = {𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))} | |
2 | nfcv 2892 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
3 | nfwlim.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
4 | nfwlim.1 | . . . . . 6 ⊢ Ⅎ𝑥𝑅 | |
5 | 3, 3, 4 | nfinf 9525 | . . . . 5 ⊢ Ⅎ𝑥inf(𝐴, 𝐴, 𝑅) |
6 | 2, 5 | nfne 3033 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅) |
7 | 4, 3, 2 | nfpred 6317 | . . . . . 6 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑦) |
8 | 7, 3, 4 | nfsup 9494 | . . . . 5 ⊢ Ⅎ𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅) |
9 | 8 | nfeq2 2910 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅) |
10 | 6, 9 | nfan 1895 | . . 3 ⊢ Ⅎ𝑥(𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)) |
11 | 10, 3 | nfrabw 3457 | . 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))} |
12 | 1, 11 | nfcxfr 2890 | 1 ⊢ Ⅎ𝑥WLim(𝑅, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1534 Ⅎwnfc 2876 ≠ wne 2930 {crab 3419 Predcpred 6311 supcsup 9483 infcinf 9484 WLimcwlim 35635 |
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 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-xp 5688 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-sup 9485 df-inf 9486 df-wlim 35637 |
This theorem is referenced by: (None) |
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