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Theorem bnj1449 31454
Description: Technical lemma for bnj60 31468. 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.)
Hypotheses
Ref Expression
bnj1449.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1449.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1449.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1449.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1449.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1449.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1449.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 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, (𝐺𝑍)⟩})
bnj1449.13 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1449.14 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1449.15 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
bnj1449.16 (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
bnj1449.17 (𝜃 ↔ (𝜒𝑧𝐸))
bnj1449.18 (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))
bnj1449.19 (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj1449 (𝜁 → ∀𝑓𝜁)
Distinct variable groups:   𝐴,𝑓   𝑓,𝐸   𝑅,𝑓   𝑥,𝑓   𝑦,𝑓   𝑧,𝑓
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜂(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜁(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑥,𝑦,𝑧,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑅(𝑥,𝑦,𝑧,𝑑)   𝐸(𝑥,𝑦,𝑧,𝑑)   𝐺(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑊(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑧,𝑓,𝑑)

Proof of Theorem bnj1449
StepHypRef Expression
1 bnj1449.19 . . 3 (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
2 bnj1449.17 . . . . 5 (𝜃 ↔ (𝜒𝑧𝐸))
3 bnj1449.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
4 bnj1449.6 . . . . . . . . 9 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
5 nfv 1995 . . . . . . . . . 10 𝑓 𝑅 FrSe 𝐴
6 bnj1449.5 . . . . . . . . . . . 12 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
7 nfe1 2183 . . . . . . . . . . . . . 14 𝑓𝑓𝜏
87nfn 1935 . . . . . . . . . . . . 13 𝑓 ¬ ∃𝑓𝜏
9 nfcv 2913 . . . . . . . . . . . . 13 𝑓𝐴
108, 9nfrab 3272 . . . . . . . . . . . 12 𝑓{𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
116, 10nfcxfr 2911 . . . . . . . . . . 11 𝑓𝐷
12 nfcv 2913 . . . . . . . . . . 11 𝑓
1311, 12nfne 3043 . . . . . . . . . 10 𝑓 𝐷 ≠ ∅
145, 13nfan 1980 . . . . . . . . 9 𝑓(𝑅 FrSe 𝐴𝐷 ≠ ∅)
154, 14nfxfr 1929 . . . . . . . 8 𝑓𝜓
1611nfcri 2907 . . . . . . . 8 𝑓 𝑥𝐷
17 nfv 1995 . . . . . . . . 9 𝑓 ¬ 𝑦𝑅𝑥
1811, 17nfral 3094 . . . . . . . 8 𝑓𝑦𝐷 ¬ 𝑦𝑅𝑥
1915, 16, 18nf3an 1983 . . . . . . 7 𝑓(𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
203, 19nfxfr 1929 . . . . . 6 𝑓𝜒
21 nfv 1995 . . . . . 6 𝑓 𝑧𝐸
2220, 21nfan 1980 . . . . 5 𝑓(𝜒𝑧𝐸)
232, 22nfxfr 1929 . . . 4 𝑓𝜃
24 nfv 1995 . . . 4 𝑓 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)
2523, 24nfan 1980 . . 3 𝑓(𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
261, 25nfxfr 1929 . 2 𝑓𝜁
2726nf5ri 2219 1 (𝜁 → ∀𝑓𝜁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071  wal 1629   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  wral 3061  wrex 3062  {crab 3065  [wsbc 3587  cun 3721  wss 3723  c0 4063  {csn 4317  cop 4323   cuni 4575   class class class wbr 4787  dom cdm 5250  cres 5252   Fn wfn 6025  cfv 6030   predc-bnj14 31094   FrSe w-bnj15 31098   trClc-bnj18 31100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rab 3070
This theorem is referenced by:  bnj1450  31456
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