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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1444 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 31130. 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 |
---|---|
bnj1444.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1444.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1444.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1444.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
bnj1444.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} |
bnj1444.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) |
bnj1444.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) |
bnj1444.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) |
bnj1444.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
bnj1444.10 | ⊢ 𝑃 = ∪ 𝐻 |
bnj1444.11 | ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1444.12 | ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
bnj1444.13 | ⊢ 𝑊 = 〈𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))〉 |
bnj1444.14 | ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) |
bnj1444.15 | ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅)) |
bnj1444.16 | ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) |
bnj1444.17 | ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸)) |
bnj1444.18 | ⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥})) |
bnj1444.19 | ⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))) |
bnj1444.20 | ⊢ (𝜌 ↔ (𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓)) |
Ref | Expression |
---|---|
bnj1444 | ⊢ (𝜌 → ∀𝑦𝜌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1444.20 | . . 3 ⊢ (𝜌 ↔ (𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓)) | |
2 | bnj1444.19 | . . . . 5 ⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))) | |
3 | bnj1444.17 | . . . . . . 7 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸)) | |
4 | bnj1444.7 | . . . . . . . . 9 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) | |
5 | nfv 1843 | . . . . . . . . . 10 ⊢ Ⅎ𝑦𝜓 | |
6 | nfv 1843 | . . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐷 | |
7 | nfra1 2941 | . . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥 | |
8 | 5, 6, 7 | nf3an 1831 | . . . . . . . . 9 ⊢ Ⅎ𝑦(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥) |
9 | 4, 8 | nfxfr 1779 | . . . . . . . 8 ⊢ Ⅎ𝑦𝜒 |
10 | nfv 1843 | . . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐸 | |
11 | 9, 10 | nfan 1828 | . . . . . . 7 ⊢ Ⅎ𝑦(𝜒 ∧ 𝑧 ∈ 𝐸) |
12 | 3, 11 | nfxfr 1779 | . . . . . 6 ⊢ Ⅎ𝑦𝜃 |
13 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅) | |
14 | 12, 13 | nfan 1828 | . . . . 5 ⊢ Ⅎ𝑦(𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) |
15 | 2, 14 | nfxfr 1779 | . . . 4 ⊢ Ⅎ𝑦𝜁 |
16 | bnj1444.9 | . . . . . 6 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | |
17 | nfre1 3005 | . . . . . . 7 ⊢ Ⅎ𝑦∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ | |
18 | 17 | nfab 2769 | . . . . . 6 ⊢ Ⅎ𝑦{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
19 | 16, 18 | nfcxfr 2762 | . . . . 5 ⊢ Ⅎ𝑦𝐻 |
20 | 19 | nfcri 2758 | . . . 4 ⊢ Ⅎ𝑦 𝑓 ∈ 𝐻 |
21 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ dom 𝑓 | |
22 | 15, 20, 21 | nf3an 1831 | . . 3 ⊢ Ⅎ𝑦(𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓) |
23 | 1, 22 | nfxfr 1779 | . 2 ⊢ Ⅎ𝑦𝜌 |
24 | 23 | nf5ri 2065 | 1 ⊢ (𝜌 → ∀𝑦𝜌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∀wal 1481 = wceq 1483 ∃wex 1704 ∈ wcel 1990 {cab 2608 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 [wsbc 3435 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 {csn 4177 〈cop 4183 ∪ cuni 4436 class class class wbr 4653 dom cdm 5114 ↾ cres 5116 Fn wfn 5883 ‘cfv 5888 predc-bnj14 30754 FrSe w-bnj15 30758 trClc-bnj18 30760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 |
This theorem is referenced by: bnj1450 31118 |
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