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Theorem bnj1467 31977
Description: Technical lemma for bnj60 31985. 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
bnj1467.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1467.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1467.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1467.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1467.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1467.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1467.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1467.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1467.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1467.10 𝑃 = 𝐻
bnj1467.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1467.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
Assertion
Ref Expression
bnj1467 (𝑤𝑄 → ∀𝑑 𝑤𝑄)
Distinct variable groups:   𝐴,𝑑,𝑤,𝑥   𝐵,𝑓   𝑤,𝐶   𝐺,𝑑,𝑤   𝑤,𝐻   𝑤,𝑃   𝑅,𝑑,𝑤,𝑥   𝑤,𝑍   𝑓,𝑑,𝑤,𝑥   𝑦,𝑑,𝑥
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑤,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑤,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑤,𝑓,𝑑)   𝐴(𝑦,𝑓)   𝐵(𝑥,𝑦,𝑤,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑤,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑤,𝑓,𝑑)   𝑅(𝑦,𝑓)   𝐺(𝑥,𝑦,𝑓)   𝐻(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑤,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑤,𝑓,𝑑)

Proof of Theorem bnj1467
StepHypRef Expression
1 bnj1467.12 . . 3 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
2 bnj1467.10 . . . . 5 𝑃 = 𝐻
3 bnj1467.9 . . . . . . 7 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
4 nfcv 2932 . . . . . . . . 9 𝑑 pred(𝑥, 𝐴, 𝑅)
5 bnj1467.8 . . . . . . . . . 10 (𝜏′[𝑦 / 𝑥]𝜏)
6 nfcv 2932 . . . . . . . . . . 11 𝑑𝑦
7 bnj1467.4 . . . . . . . . . . . 12 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
8 bnj1467.3 . . . . . . . . . . . . . . 15 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
9 nfre1 3251 . . . . . . . . . . . . . . . 16 𝑑𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
109nfab 2938 . . . . . . . . . . . . . . 15 𝑑{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
118, 10nfcxfr 2930 . . . . . . . . . . . . . 14 𝑑𝐶
1211nfcri 2926 . . . . . . . . . . . . 13 𝑑 𝑓𝐶
13 nfv 1873 . . . . . . . . . . . . 13 𝑑dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
1412, 13nfan 1862 . . . . . . . . . . . 12 𝑑(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
157, 14nfxfr 1815 . . . . . . . . . . 11 𝑑𝜏
166, 15nfsbc 3703 . . . . . . . . . 10 𝑑[𝑦 / 𝑥]𝜏
175, 16nfxfr 1815 . . . . . . . . 9 𝑑𝜏′
184, 17nfrex 3253 . . . . . . . 8 𝑑𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
1918nfab 2938 . . . . . . 7 𝑑{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
203, 19nfcxfr 2930 . . . . . 6 𝑑𝐻
2120nfuni 4718 . . . . 5 𝑑 𝐻
222, 21nfcxfr 2930 . . . 4 𝑑𝑃
23 nfcv 2932 . . . . . 6 𝑑𝑥
24 nfcv 2932 . . . . . . 7 𝑑𝐺
25 bnj1467.11 . . . . . . . 8 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
2622, 4nfres 5697 . . . . . . . . 9 𝑑(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
2723, 26nfop 4693 . . . . . . . 8 𝑑𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
2825, 27nfcxfr 2930 . . . . . . 7 𝑑𝑍
2924, 28nffv 6509 . . . . . 6 𝑑(𝐺𝑍)
3023, 29nfop 4693 . . . . 5 𝑑𝑥, (𝐺𝑍)⟩
3130nfsn 4517 . . . 4 𝑑{⟨𝑥, (𝐺𝑍)⟩}
3222, 31nfun 4030 . . 3 𝑑(𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
331, 32nfcxfr 2930 . 2 𝑑𝑄
3433nfcrii 2925 1 (𝑤𝑄 → ∀𝑑 𝑤𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  w3a 1068  wal 1505   = wceq 1507  wex 1742  wcel 2050  {cab 2758  wne 2967  wral 3088  wrex 3089  {crab 3092  [wsbc 3681  cun 3827  wss 3829  c0 4178  {csn 4441  cop 4447   cuni 4712   class class class wbr 4929  dom cdm 5407  cres 5409   Fn wfn 6183  cfv 6188   predc-bnj14 31612   FrSe w-bnj15 31616   trClc-bnj18 31618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-xp 5413  df-res 5419  df-iota 6152  df-fv 6196
This theorem is referenced by:  bnj1463  31978
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