Mathbox for Jonathan Ben-Naim < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1388 Structured version   Visualization version   GIF version

Theorem bnj1388 30355
 Description: Technical lemma for bnj60 30384. 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 1830 . . . 4 𝑦𝜓
3 nfv 1830 . . . 4 𝑦 𝑥𝐷
4 nfra1 2925 . . . 4 𝑦𝑦𝐷 ¬ 𝑦𝑅𝑥
52, 3, 4nf3an 1819 . . 3 𝑦(𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
61, 5nfxfr 1771 . 2 𝑦𝜒
7 bnj1152 30320 . . . . . 6 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ↔ (𝑦𝐴𝑦𝑅𝑥))
87simplbi 475 . . . . 5 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝐴)
98adantl 481 . . . 4 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑦𝐴)
107biimpi 205 . . . . . . . . 9 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → (𝑦𝐴𝑦𝑅𝑥))
1110adantl 481 . . . . . . . 8 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑦𝐴𝑦𝑅𝑥))
1211simprd 478 . . . . . . 7 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑦𝑅𝑥)
131simp3bi 1071 . . . . . . . 8 (𝜒 → ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
1413adantr 480 . . . . . . 7 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
15 df-ral 2901 . . . . . . . . 9 (∀𝑦𝐷 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦(𝑦𝐷 → ¬ 𝑦𝑅𝑥))
16 con2b 348 . . . . . . . . . 10 ((𝑦𝐷 → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦𝐷))
1716albii 1737 . . . . . . . . 9 (∀𝑦(𝑦𝐷 → ¬ 𝑦𝑅𝑥) ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦𝐷))
1815, 17bitri 263 . . . . . . . 8 (∀𝑦𝐷 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦𝐷))
19 sp 2041 . . . . . . . . 9 (∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦𝐷) → (𝑦𝑅𝑥 → ¬ 𝑦𝐷))
2019impcom 445 . . . . . . . 8 ((𝑦𝑅𝑥 ∧ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦𝐷)) → ¬ 𝑦𝐷)
2118, 20sylan2b 491 . . . . . . 7 ((𝑦𝑅𝑥 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥) → ¬ 𝑦𝐷)
2212, 14, 21syl2anc 691 . . . . . 6 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑦𝐷)
23 bnj1388.5 . . . . . . . 8 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2423eleq2i 2680 . . . . . . 7 (𝑦𝐷𝑦 ∈ {𝑥𝐴 ∣ ¬ ∃𝑓𝜏})
25 nfcv 2751 . . . . . . . 8 𝑥𝑦
26 nfcv 2751 . . . . . . . 8 𝑥𝐴
27 bnj1388.8 . . . . . . . . . . 11 (𝜏′[𝑦 / 𝑥]𝜏)
28 nfsbc1v 3422 . . . . . . . . . . 11 𝑥[𝑦 / 𝑥]𝜏
2927, 28nfxfr 1771 . . . . . . . . . 10 𝑥𝜏′
3029nfex 2140 . . . . . . . . 9 𝑥𝑓𝜏′
3130nfn 1768 . . . . . . . 8 𝑥 ¬ ∃𝑓𝜏′
32 sbceq1a 3413 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝜏[𝑦 / 𝑥]𝜏))
3332, 27syl6bbr 277 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜏𝜏′))
3433exbidv 1837 . . . . . . . . 9 (𝑥 = 𝑦 → (∃𝑓𝜏 ↔ ∃𝑓𝜏′))
3534notbid 307 . . . . . . . 8 (𝑥 = 𝑦 → (¬ ∃𝑓𝜏 ↔ ¬ ∃𝑓𝜏′))
3625, 26, 31, 35elrabf 3329 . . . . . . 7 (𝑦 ∈ {𝑥𝐴 ∣ ¬ ∃𝑓𝜏} ↔ (𝑦𝐴 ∧ ¬ ∃𝑓𝜏′))
3724, 36bitri 263 . . . . . 6 (𝑦𝐷 ↔ (𝑦𝐴 ∧ ¬ ∃𝑓𝜏′))
3822, 37sylnib 317 . . . . 5 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ (𝑦𝐴 ∧ ¬ ∃𝑓𝜏′))
39 iman 439 . . . . 5 ((𝑦𝐴 → ∃𝑓𝜏′) ↔ ¬ (𝑦𝐴 ∧ ¬ ∃𝑓𝜏′))
4038, 39sylibr 223 . . . 4 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑦𝐴 → ∃𝑓𝜏′))
419, 40mpd 15 . . 3 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃𝑓𝜏′)
4241ex 449 . 2 (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃𝑓𝜏′))
436, 42ralrimi 2940 1 (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  [wsbc 3402   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583  dom cdm 5038   ↾ cres 5040   Fn wfn 5799  ‘cfv 5804   predc-bnj14 30007   FrSe w-bnj15 30011   trClc-bnj18 30013 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-bnj14 30008 This theorem is referenced by:  bnj1398  30356  bnj1489  30378
 Copyright terms: Public domain W3C validator