Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1466 Structured version   Visualization version   GIF version

Theorem bnj1466 32320
Description: Technical lemma for bnj60 32329. 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 32088 . . . . . . 7 (𝑤𝐻 → ∀𝑓 𝑤𝐻)
54nfcii 2965 . . . . . 6 𝑓𝐻
65nfuni 4838 . . . . 5 𝑓 𝐻
72, 6nfcxfr 2975 . . . 4 𝑓𝑃
8 nfcv 2977 . . . . . 6 𝑓𝑥
9 nfcv 2977 . . . . . . 7 𝑓𝐺
10 bnj1466.11 . . . . . . . 8 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
11 nfcv 2977 . . . . . . . . . 10 𝑓 pred(𝑥, 𝐴, 𝑅)
127, 11nfres 5849 . . . . . . . . 9 𝑓(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
138, 12nfop 4812 . . . . . . . 8 𝑓𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
1410, 13nfcxfr 2975 . . . . . . 7 𝑓𝑍
159, 14nffv 6674 . . . . . 6 𝑓(𝐺𝑍)
168, 15nfop 4812 . . . . 5 𝑓𝑥, (𝐺𝑍)⟩
1716nfsn 4636 . . . 4 𝑓{⟨𝑥, (𝐺𝑍)⟩}
187, 17nfun 4140 . . 3 𝑓(𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
191, 18nfcxfr 2975 . 2 𝑓𝑄
2019nfcrii 2970 1 (𝑤𝑄 → ∀𝑓 𝑤𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083  wal 1531   = wceq 1533  wex 1776  wcel 2110  {cab 2799  wne 3016  wral 3138  wrex 3139  {crab 3142  [wsbc 3771  cun 3933  wss 3935  c0 4290  {csn 4560  cop 4566   cuni 4831   class class class wbr 5058  dom cdm 5549  cres 5551   Fn wfn 6344  cfv 6349   predc-bnj14 31953   FrSe w-bnj15 31957   trClc-bnj18 31959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-xp 5555  df-res 5561  df-iota 6308  df-fv 6357
This theorem is referenced by:  bnj1463  32322  bnj1491  32324
  Copyright terms: Public domain W3C validator