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Theorem bnj1444 31237
Description: Technical lemma for bnj60 31256. 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 1883 . . . . . . . . . 10 𝑦𝜓
6 nfv 1883 . . . . . . . . . 10 𝑦 𝑥𝐷
7 nfra1 2970 . . . . . . . . . 10 𝑦𝑦𝐷 ¬ 𝑦𝑅𝑥
85, 6, 7nf3an 1871 . . . . . . . . 9 𝑦(𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
94, 8nfxfr 1819 . . . . . . . 8 𝑦𝜒
10 nfv 1883 . . . . . . . 8 𝑦 𝑧𝐸
119, 10nfan 1868 . . . . . . 7 𝑦(𝜒𝑧𝐸)
123, 11nfxfr 1819 . . . . . 6 𝑦𝜃
13 nfv 1883 . . . . . 6 𝑦 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)
1412, 13nfan 1868 . . . . 5 𝑦(𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
152, 14nfxfr 1819 . . . 4 𝑦𝜁
16 bnj1444.9 . . . . . 6 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
17 nfre1 3034 . . . . . . 7 𝑦𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
1817nfab 2798 . . . . . 6 𝑦{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
1916, 18nfcxfr 2791 . . . . 5 𝑦𝐻
2019nfcri 2787 . . . 4 𝑦 𝑓𝐻
21 nfv 1883 . . . 4 𝑦 𝑧 ∈ dom 𝑓
2215, 20, 21nf3an 1871 . . 3 𝑦(𝜁𝑓𝐻𝑧 ∈ dom 𝑓)
231, 22nfxfr 1819 . 2 𝑦𝜌
2423nf5ri 2103 1 (𝜌 → ∀𝑦𝜌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  {crab 2945  [wsbc 3468  cun 3605  wss 3607  c0 3948  {csn 4210  cop 4216   cuni 4468   class class class wbr 4685  dom cdm 5143  cres 5145   Fn wfn 5921  cfv 5926   predc-bnj14 30882   FrSe w-bnj15 30886   trClc-bnj18 30888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947
This theorem is referenced by:  bnj1450  31244
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