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Theorem bnj1388 31433
Description: Technical lemma for bnj60 31462. 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 1994 . . . 4 𝑦𝜓
3 nfv 1994 . . . 4 𝑦 𝑥𝐷
4 nfra1 3089 . . . 4 𝑦𝑦𝐷 ¬ 𝑦𝑅𝑥
52, 3, 4nf3an 1982 . . 3 𝑦(𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
61, 5nfxfr 1928 . 2 𝑦𝜒
7 bnj1152 31398 . . . . . 6 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ↔ (𝑦𝐴𝑦𝑅𝑥))
87simplbi 479 . . . . 5 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝐴)
98adantl 467 . . . 4 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑦𝐴)
107biimpi 206 . . . . . . . . 9 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → (𝑦𝐴𝑦𝑅𝑥))
1110adantl 467 . . . . . . . 8 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑦𝐴𝑦𝑅𝑥))
1211simprd 477 . . . . . . 7 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑦𝑅𝑥)
131simp3bi 1140 . . . . . . . 8 (𝜒 → ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
1413adantr 466 . . . . . . 7 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
15 df-ral 3065 . . . . . . . . 9 (∀𝑦𝐷 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦(𝑦𝐷 → ¬ 𝑦𝑅𝑥))
16 con2b 348 . . . . . . . . . 10 ((𝑦𝐷 → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦𝐷))
1716albii 1894 . . . . . . . . 9 (∀𝑦(𝑦𝐷 → ¬ 𝑦𝑅𝑥) ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦𝐷))
1815, 17bitri 264 . . . . . . . 8 (∀𝑦𝐷 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦𝐷))
19 sp 2206 . . . . . . . . 9 (∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦𝐷) → (𝑦𝑅𝑥 → ¬ 𝑦𝐷))
2019impcom 394 . . . . . . . 8 ((𝑦𝑅𝑥 ∧ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦𝐷)) → ¬ 𝑦𝐷)
2118, 20sylan2b 573 . . . . . . 7 ((𝑦𝑅𝑥 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥) → ¬ 𝑦𝐷)
2212, 14, 21syl2anc 565 . . . . . 6 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑦𝐷)
23 bnj1388.5 . . . . . . . 8 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2423eleq2i 2841 . . . . . . 7 (𝑦𝐷𝑦 ∈ {𝑥𝐴 ∣ ¬ ∃𝑓𝜏})
25 nfcv 2912 . . . . . . . 8 𝑥𝑦
26 nfcv 2912 . . . . . . . 8 𝑥𝐴
27 bnj1388.8 . . . . . . . . . . 11 (𝜏′[𝑦 / 𝑥]𝜏)
28 nfsbc1v 3605 . . . . . . . . . . 11 𝑥[𝑦 / 𝑥]𝜏
2927, 28nfxfr 1928 . . . . . . . . . 10 𝑥𝜏′
3029nfex 2317 . . . . . . . . 9 𝑥𝑓𝜏′
3130nfn 1934 . . . . . . . 8 𝑥 ¬ ∃𝑓𝜏′
32 sbceq1a 3596 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝜏[𝑦 / 𝑥]𝜏))
3332, 27syl6bbr 278 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜏𝜏′))
3433exbidv 2001 . . . . . . . . 9 (𝑥 = 𝑦 → (∃𝑓𝜏 ↔ ∃𝑓𝜏′))
3534notbid 307 . . . . . . . 8 (𝑥 = 𝑦 → (¬ ∃𝑓𝜏 ↔ ¬ ∃𝑓𝜏′))
3625, 26, 31, 35elrabf 3509 . . . . . . 7 (𝑦 ∈ {𝑥𝐴 ∣ ¬ ∃𝑓𝜏} ↔ (𝑦𝐴 ∧ ¬ ∃𝑓𝜏′))
3724, 36bitri 264 . . . . . 6 (𝑦𝐷 ↔ (𝑦𝐴 ∧ ¬ ∃𝑓𝜏′))
3822, 37sylnib 317 . . . . 5 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ (𝑦𝐴 ∧ ¬ ∃𝑓𝜏′))
39 iman 388 . . . . 5 ((𝑦𝐴 → ∃𝑓𝜏′) ↔ ¬ (𝑦𝐴 ∧ ¬ ∃𝑓𝜏′))
4038, 39sylibr 224 . . . 4 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑦𝐴 → ∃𝑓𝜏′))
419, 40mpd 15 . . 3 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃𝑓𝜏′)
4241ex 397 . 2 (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃𝑓𝜏′))
436, 42ralrimi 3105 1 (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1070  wal 1628   = wceq 1630  wex 1851  wcel 2144  {cab 2756  wne 2942  wral 3060  wrex 3061  {crab 3064  [wsbc 3585  cun 3719  wss 3721  c0 4061  {csn 4314  cop 4320   class class class wbr 4784  dom cdm 5249  cres 5251   Fn wfn 6026  cfv 6031   predc-bnj14 31088   FrSe w-bnj15 31092   trClc-bnj18 31094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-bnj14 31089
This theorem is referenced by:  bnj1398  31434  bnj1489  31456
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