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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1449 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32329. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1449.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1449.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1449.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1449.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
bnj1449.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} |
bnj1449.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) |
bnj1449.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) |
bnj1449.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) |
bnj1449.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
bnj1449.10 | ⊢ 𝑃 = ∪ 𝐻 |
bnj1449.11 | ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1449.12 | ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
bnj1449.13 | ⊢ 𝑊 = 〈𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))〉 |
bnj1449.14 | ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) |
bnj1449.15 | ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅)) |
bnj1449.16 | ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) |
bnj1449.17 | ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸)) |
bnj1449.18 | ⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥})) |
bnj1449.19 | ⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))) |
Ref | Expression |
---|---|
bnj1449 | ⊢ (𝜁 → ∀𝑓𝜁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1449.19 | . . 3 ⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))) | |
2 | bnj1449.17 | . . . . 5 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸)) | |
3 | bnj1449.7 | . . . . . . 7 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) | |
4 | bnj1449.6 | . . . . . . . . 9 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) | |
5 | nfv 1911 | . . . . . . . . . 10 ⊢ Ⅎ𝑓 𝑅 FrSe 𝐴 | |
6 | bnj1449.5 | . . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} | |
7 | nfe1 2150 | . . . . . . . . . . . . . 14 ⊢ Ⅎ𝑓∃𝑓𝜏 | |
8 | 7 | nfn 1853 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑓 ¬ ∃𝑓𝜏 |
9 | nfcv 2977 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑓𝐴 | |
10 | 8, 9 | nfrabw 3385 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑓{𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} |
11 | 6, 10 | nfcxfr 2975 | . . . . . . . . . . 11 ⊢ Ⅎ𝑓𝐷 |
12 | nfcv 2977 | . . . . . . . . . . 11 ⊢ Ⅎ𝑓∅ | |
13 | 11, 12 | nfne 3119 | . . . . . . . . . 10 ⊢ Ⅎ𝑓 𝐷 ≠ ∅ |
14 | 5, 13 | nfan 1896 | . . . . . . . . 9 ⊢ Ⅎ𝑓(𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅) |
15 | 4, 14 | nfxfr 1849 | . . . . . . . 8 ⊢ Ⅎ𝑓𝜓 |
16 | 11 | nfcri 2971 | . . . . . . . 8 ⊢ Ⅎ𝑓 𝑥 ∈ 𝐷 |
17 | nfv 1911 | . . . . . . . . 9 ⊢ Ⅎ𝑓 ¬ 𝑦𝑅𝑥 | |
18 | 11, 17 | nfralw 3225 | . . . . . . . 8 ⊢ Ⅎ𝑓∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥 |
19 | 15, 16, 18 | nf3an 1898 | . . . . . . 7 ⊢ Ⅎ𝑓(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥) |
20 | 3, 19 | nfxfr 1849 | . . . . . 6 ⊢ Ⅎ𝑓𝜒 |
21 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑓 𝑧 ∈ 𝐸 | |
22 | 20, 21 | nfan 1896 | . . . . 5 ⊢ Ⅎ𝑓(𝜒 ∧ 𝑧 ∈ 𝐸) |
23 | 2, 22 | nfxfr 1849 | . . . 4 ⊢ Ⅎ𝑓𝜃 |
24 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑓 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅) | |
25 | 23, 24 | nfan 1896 | . . 3 ⊢ Ⅎ𝑓(𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) |
26 | 1, 25 | nfxfr 1849 | . 2 ⊢ Ⅎ𝑓𝜁 |
27 | 26 | nf5ri 2191 | 1 ⊢ (𝜁 → ∀𝑓𝜁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∀wal 1531 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 [wsbc 3771 ∪ cun 3933 ⊆ wss 3935 ∅c0 4290 {csn 4560 〈cop 4566 ∪ cuni 4831 class class class wbr 5058 dom cdm 5549 ↾ cres 5551 Fn wfn 6344 ‘cfv 6349 predc-bnj14 31953 FrSe w-bnj15 31957 trClc-bnj18 31959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 |
This theorem is referenced by: bnj1450 32317 |
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