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Theorem bnj1388 34044
Description: Technical lemma for bnj60 34073. 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
bnj1388.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1388.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1388.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1388.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1388.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1388.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1388.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1388.8 (𝜏′[𝑦 / 𝑥]𝜏)
Assertion
Ref Expression
bnj1388 (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
Distinct variable groups:   𝑥,𝐴,𝑦   𝐵,𝑓   𝑦,𝐷   𝑥,𝑅,𝑦   𝑓,𝑑,𝑥   𝑦,𝑓   𝜓,𝑦   𝜏,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑓,𝑑)   𝐴(𝑓,𝑑)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑑)   𝐷(𝑥,𝑓,𝑑)   𝑅(𝑓,𝑑)   𝐺(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1388
StepHypRef Expression
1 bnj1388.7 . . 3 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
2 nfv 1918 . . . 4 𝑦𝜓
3 nfv 1918 . . . 4 𝑦 𝑥𝐷
4 nfra1 3282 . . . 4 𝑦𝑦𝐷 ¬ 𝑦𝑅𝑥
52, 3, 4nf3an 1905 . . 3 𝑦(𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
61, 5nfxfr 1856 . 2 𝑦𝜒
7 bnj1152 34009 . . . . . 6 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ↔ (𝑦𝐴𝑦𝑅𝑥))
87simplbi 499 . . . . 5 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝐴)
98adantl 483 . . . 4 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑦𝐴)
107biimpi 215 . . . . . . . . 9 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → (𝑦𝐴𝑦𝑅𝑥))
1110adantl 483 . . . . . . . 8 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑦𝐴𝑦𝑅𝑥))
1211simprd 497 . . . . . . 7 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑦𝑅𝑥)
131simp3bi 1148 . . . . . . . 8 (𝜒 → ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
1413adantr 482 . . . . . . 7 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
15 df-ral 3063 . . . . . . . . 9 (∀𝑦𝐷 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦(𝑦𝐷 → ¬ 𝑦𝑅𝑥))
16 con2b 360 . . . . . . . . . 10 ((𝑦𝐷 → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦𝐷))
1716albii 1822 . . . . . . . . 9 (∀𝑦(𝑦𝐷 → ¬ 𝑦𝑅𝑥) ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦𝐷))
1815, 17bitri 275 . . . . . . . 8 (∀𝑦𝐷 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦𝐷))
19 sp 2177 . . . . . . . . 9 (∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦𝐷) → (𝑦𝑅𝑥 → ¬ 𝑦𝐷))
2019impcom 409 . . . . . . . 8 ((𝑦𝑅𝑥 ∧ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦𝐷)) → ¬ 𝑦𝐷)
2118, 20sylan2b 595 . . . . . . 7 ((𝑦𝑅𝑥 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥) → ¬ 𝑦𝐷)
2212, 14, 21syl2anc 585 . . . . . 6 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑦𝐷)
23 bnj1388.5 . . . . . . . 8 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2423eleq2i 2826 . . . . . . 7 (𝑦𝐷𝑦 ∈ {𝑥𝐴 ∣ ¬ ∃𝑓𝜏})
25 nfcv 2904 . . . . . . . 8 𝑥𝑦
26 nfcv 2904 . . . . . . . 8 𝑥𝐴
27 bnj1388.8 . . . . . . . . . . 11 (𝜏′[𝑦 / 𝑥]𝜏)
28 nfsbc1v 3798 . . . . . . . . . . 11 𝑥[𝑦 / 𝑥]𝜏
2927, 28nfxfr 1856 . . . . . . . . . 10 𝑥𝜏′
3029nfex 2318 . . . . . . . . 9 𝑥𝑓𝜏′
3130nfn 1861 . . . . . . . 8 𝑥 ¬ ∃𝑓𝜏′
32 sbceq1a 3789 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝜏[𝑦 / 𝑥]𝜏))
3332, 27bitr4di 289 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜏𝜏′))
3433exbidv 1925 . . . . . . . . 9 (𝑥 = 𝑦 → (∃𝑓𝜏 ↔ ∃𝑓𝜏′))
3534notbid 318 . . . . . . . 8 (𝑥 = 𝑦 → (¬ ∃𝑓𝜏 ↔ ¬ ∃𝑓𝜏′))
3625, 26, 31, 35elrabf 3680 . . . . . . 7 (𝑦 ∈ {𝑥𝐴 ∣ ¬ ∃𝑓𝜏} ↔ (𝑦𝐴 ∧ ¬ ∃𝑓𝜏′))
3724, 36bitri 275 . . . . . 6 (𝑦𝐷 ↔ (𝑦𝐴 ∧ ¬ ∃𝑓𝜏′))
3822, 37sylnib 328 . . . . 5 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ (𝑦𝐴 ∧ ¬ ∃𝑓𝜏′))
39 iman 403 . . . . 5 ((𝑦𝐴 → ∃𝑓𝜏′) ↔ ¬ (𝑦𝐴 ∧ ¬ ∃𝑓𝜏′))
4038, 39sylibr 233 . . . 4 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑦𝐴 → ∃𝑓𝜏′))
419, 40mpd 15 . . 3 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃𝑓𝜏′)
4241ex 414 . 2 (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃𝑓𝜏′))
436, 42ralrimi 3255 1 (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wne 2941  wral 3062  wrex 3071  {crab 3433  [wsbc 3778  cun 3947  wss 3949  c0 4323  {csn 4629  cop 4635   class class class wbr 5149  dom cdm 5677  cres 5679   Fn wfn 6539  cfv 6544   predc-bnj14 33699   FrSe w-bnj15 33703   trClc-bnj18 33705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-bnj14 33700
This theorem is referenced by:  bnj1398  34045  bnj1489  34067
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