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Theorem bnj1444 32218
 Description: Technical lemma for bnj60 32237. 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
bnj1444.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1444.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1444.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1444.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1444.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1444.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1444.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 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, (𝐺𝑍)⟩})
bnj1444.13 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1444.14 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1444.15 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
bnj1444.16 (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
bnj1444.17 (𝜃 ↔ (𝜒𝑧𝐸))
bnj1444.18 (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))
bnj1444.19 (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
bnj1444.20 (𝜌 ↔ (𝜁𝑓𝐻𝑧 ∈ dom 𝑓))
Assertion
Ref Expression
bnj1444 (𝜌 → ∀𝑦𝜌)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐷   𝑦,𝐸   𝑦,𝑅   𝑦,𝑓   𝜓,𝑦   𝑥,𝑦   𝑦,𝑧
Allowed substitution hints:   𝜓(𝑥,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)   𝜌(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑥,𝑧,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑅(𝑥,𝑧,𝑓,𝑑)   𝐸(𝑥,𝑧,𝑓,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 bnj1444
StepHypRef Expression
1 bnj1444.20 . . 3 (𝜌 ↔ (𝜁𝑓𝐻𝑧 ∈ dom 𝑓))
2 bnj1444.19 . . . . 5 (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
3 bnj1444.17 . . . . . . 7 (𝜃 ↔ (𝜒𝑧𝐸))
4 bnj1444.7 . . . . . . . . 9 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
5 nfv 1908 . . . . . . . . . 10 𝑦𝜓
6 nfv 1908 . . . . . . . . . 10 𝑦 𝑥𝐷
7 nfra1 3224 . . . . . . . . . 10 𝑦𝑦𝐷 ¬ 𝑦𝑅𝑥
85, 6, 7nf3an 1895 . . . . . . . . 9 𝑦(𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
94, 8nfxfr 1846 . . . . . . . 8 𝑦𝜒
10 nfv 1908 . . . . . . . 8 𝑦 𝑧𝐸
119, 10nfan 1893 . . . . . . 7 𝑦(𝜒𝑧𝐸)
123, 11nfxfr 1846 . . . . . 6 𝑦𝜃
13 nfv 1908 . . . . . 6 𝑦 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)
1412, 13nfan 1893 . . . . 5 𝑦(𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
152, 14nfxfr 1846 . . . 4 𝑦𝜁
16 bnj1444.9 . . . . . 6 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
17 nfre1 3311 . . . . . . 7 𝑦𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
1817nfab 2989 . . . . . 6 𝑦{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
1916, 18nfcxfr 2980 . . . . 5 𝑦𝐻
2019nfcri 2976 . . . 4 𝑦 𝑓𝐻
21 nfv 1908 . . . 4 𝑦 𝑧 ∈ dom 𝑓
2215, 20, 21nf3an 1895 . . 3 𝑦(𝜁𝑓𝐻𝑧 ∈ dom 𝑓)
231, 22nfxfr 1846 . 2 𝑦𝜌
2423nf5ri 2187 1 (𝜌 → ∀𝑦𝜌)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1081  ∀wal 1528   = wceq 1530  ∃wex 1773   ∈ wcel 2107  {cab 2804   ≠ wne 3021  ∀wral 3143  ∃wrex 3144  {crab 3147  [wsbc 3776   ∪ cun 3938   ⊆ wss 3940  ∅c0 4295  {csn 4564  ⟨cop 4570  ∪ cuni 4837   class class class wbr 5063  dom cdm 5554   ↾ cres 5556   Fn wfn 6349  ‘cfv 6354   predc-bnj14 31863   FrSe w-bnj15 31867   trClc-bnj18 31869 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149 This theorem is referenced by:  bnj1450  32225
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