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Theorem bnj1466 31450
Description: Technical lemma for bnj60 31459. 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
bnj1466.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1466.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1466.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1466.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1466.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1466.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1466.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1466.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1466.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1466.10 𝑃 = 𝐻
bnj1466.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1466.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
Assertion
Ref Expression
bnj1466 (𝑤𝑄 → ∀𝑓 𝑤𝑄)
Distinct variable groups:   𝐴,𝑓,𝑤   𝑓,𝐺,𝑤   𝑤,𝐻   𝑤,𝑃   𝑅,𝑓,𝑤   𝑤,𝑍   𝑥,𝑓,𝑤
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑤,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑤,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑤,𝑓,𝑑)   𝐴(𝑥,𝑦,𝑑)   𝐵(𝑥,𝑦,𝑤,𝑓,𝑑)   𝐶(𝑥,𝑦,𝑤,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑤,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑤,𝑓,𝑑)   𝑅(𝑥,𝑦,𝑑)   𝐺(𝑥,𝑦,𝑑)   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 bnj1466
StepHypRef Expression
1 bnj1466.12 . . 3 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
2 bnj1466.10 . . . . 5 𝑃 = 𝐻
3 bnj1466.9 . . . . . . . 8 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
43bnj1317 31221 . . . . . . 7 (𝑤𝐻 → ∀𝑓 𝑤𝐻)
54nfcii 2894 . . . . . 6 𝑓𝐻
65nfuni 4595 . . . . 5 𝑓 𝐻
72, 6nfcxfr 2901 . . . 4 𝑓𝑃
8 nfcv 2903 . . . . . 6 𝑓𝑥
9 nfcv 2903 . . . . . . 7 𝑓𝐺
10 bnj1466.11 . . . . . . . 8 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
11 nfcv 2903 . . . . . . . . . 10 𝑓 pred(𝑥, 𝐴, 𝑅)
127, 11nfres 5554 . . . . . . . . 9 𝑓(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
138, 12nfop 4570 . . . . . . . 8 𝑓𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
1410, 13nfcxfr 2901 . . . . . . 7 𝑓𝑍
159, 14nffv 6361 . . . . . 6 𝑓(𝐺𝑍)
168, 15nfop 4570 . . . . 5 𝑓𝑥, (𝐺𝑍)⟩
1716nfsn 4387 . . . 4 𝑓{⟨𝑥, (𝐺𝑍)⟩}
187, 17nfun 3913 . . 3 𝑓(𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
191, 18nfcxfr 2901 . 2 𝑓𝑄
2019nfcrii 2896 1 (𝑤𝑄 → ∀𝑓 𝑤𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072  wal 1630   = wceq 1632  wex 1853  wcel 2140  {cab 2747  wne 2933  wral 3051  wrex 3052  {crab 3055  [wsbc 3577  cun 3714  wss 3716  c0 4059  {csn 4322  cop 4328   cuni 4589   class class class wbr 4805  dom cdm 5267  cres 5269   Fn wfn 6045  cfv 6050   predc-bnj14 31085   FrSe w-bnj15 31089   trClc-bnj18 31091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-xp 5273  df-res 5279  df-iota 6013  df-fv 6058
This theorem is referenced by:  bnj1463  31452  bnj1491  31454
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