Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1467 Structured version   Visualization version   GIF version

Theorem bnj1467 32440
Description: Technical lemma for bnj60 32448. 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 2958 . . . . . . . . 9 𝑑 pred(𝑥, 𝐴, 𝑅)
5 bnj1467.8 . . . . . . . . . 10 (𝜏′[𝑦 / 𝑥]𝜏)
6 nfcv 2958 . . . . . . . . . . 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 3268 . . . . . . . . . . . . . . . 16 𝑑𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
109nfab 2964 . . . . . . . . . . . . . . 15 𝑑{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
118, 10nfcxfr 2956 . . . . . . . . . . . . . 14 𝑑𝐶
1211nfcri 2946 . . . . . . . . . . . . 13 𝑑 𝑓𝐶
13 nfv 1915 . . . . . . . . . . . . 13 𝑑dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
1412, 13nfan 1900 . . . . . . . . . . . 12 𝑑(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
157, 14nfxfr 1854 . . . . . . . . . . 11 𝑑𝜏
166, 15nfsbcw 3745 . . . . . . . . . 10 𝑑[𝑦 / 𝑥]𝜏
175, 16nfxfr 1854 . . . . . . . . 9 𝑑𝜏′
184, 17nfrex 3271 . . . . . . . 8 𝑑𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
1918nfab 2964 . . . . . . 7 𝑑{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
203, 19nfcxfr 2956 . . . . . 6 𝑑𝐻
2120nfuni 4810 . . . . 5 𝑑 𝐻
222, 21nfcxfr 2956 . . . 4 𝑑𝑃
23 nfcv 2958 . . . . . 6 𝑑𝑥
24 nfcv 2958 . . . . . . 7 𝑑𝐺
25 bnj1467.11 . . . . . . . 8 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
2622, 4nfres 5824 . . . . . . . . 9 𝑑(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
2723, 26nfop 4784 . . . . . . . 8 𝑑𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
2825, 27nfcxfr 2956 . . . . . . 7 𝑑𝑍
2924, 28nffv 6659 . . . . . 6 𝑑(𝐺𝑍)
3023, 29nfop 4784 . . . . 5 𝑑𝑥, (𝐺𝑍)⟩
3130nfsn 4606 . . . 4 𝑑{⟨𝑥, (𝐺𝑍)⟩}
3222, 31nfun 4095 . . 3 𝑑(𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
331, 32nfcxfr 2956 . 2 𝑑𝑄
3433nfcrii 2951 1 (𝑤𝑄 → ∀𝑑 𝑤𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084  wal 1536   = wceq 1538  wex 1781  wcel 2112  {cab 2779  wne 2990  wral 3109  wrex 3110  {crab 3113  [wsbc 3723  cun 3882  wss 3884  c0 4246  {csn 4528  cop 4534   cuni 4803   class class class wbr 5033  dom cdm 5523  cres 5525   Fn wfn 6323  cfv 6328   predc-bnj14 32072   FrSe w-bnj15 32076   trClc-bnj18 32078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-res 5535  df-iota 6287  df-fv 6336
This theorem is referenced by:  bnj1463  32441
  Copyright terms: Public domain W3C validator