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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 33102 | . 2 ⊢ WLim(𝑅, 𝐴) = {𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))} | |
2 | nfcv 2979 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
3 | nfwlim.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
4 | nfwlim.1 | . . . . . 6 ⊢ Ⅎ𝑥𝑅 | |
5 | 3, 3, 4 | nfinf 8948 | . . . . 5 ⊢ Ⅎ𝑥inf(𝐴, 𝐴, 𝑅) |
6 | 2, 5 | nfne 3121 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅) |
7 | 4, 3, 2 | nfpred 6155 | . . . . . 6 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑦) |
8 | 7, 3, 4 | nfsup 8917 | . . . . 5 ⊢ Ⅎ𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅) |
9 | 8 | nfeq2 2997 | . . . 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 3387 | . 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))} |
12 | 1, 11 | nfcxfr 2977 | 1 ⊢ Ⅎ𝑥WLim(𝑅, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 Ⅎwnfc 2963 ≠ wne 3018 {crab 3144 Predcpred 6149 supcsup 8906 infcinf 8907 WLimcwlim 33100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-sup 8908 df-inf 8909 df-wlim 33102 |
This theorem is referenced by: (None) |
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