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Theorem bnj1020 32264
Description: Technical lemma for bnj69 32309. 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
bnj1020.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1020.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1020.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1020.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj1020.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj1020.6 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
bnj1020.7 (𝜑′[𝑝 / 𝑛]𝜑)
bnj1020.8 (𝜓′[𝑝 / 𝑛]𝜓)
bnj1020.9 (𝜒′[𝑝 / 𝑛]𝜒)
bnj1020.10 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj1020.11 (𝜓″[𝐺 / 𝑓]𝜓′)
bnj1020.12 (𝜒″[𝐺 / 𝑓]𝜒′)
bnj1020.13 𝐷 = (ω ∖ {∅})
bnj1020.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1020.15 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj1020.16 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj1020.26 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
Assertion
Ref Expression
bnj1020 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐴,𝑝,𝑓,𝑖,𝑛,𝑦   𝐷,𝑓,𝑖,𝑛   𝑖,𝐺,𝑝   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑅,𝑝   𝑓,𝑋,𝑖,𝑛,𝑦   𝜒,𝑝   𝜂,𝑝   𝜑,𝑖   𝜃,𝑝
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝜃(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝜏(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝐴(𝑧)   𝐵(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑚,𝑝)   𝑅(𝑧)   𝐺(𝑦,𝑧,𝑓,𝑚,𝑛)   𝑋(𝑧,𝑚,𝑝)   𝜑′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj1020
StepHypRef Expression
1 bnj1019 32078 . . 3 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
2 bnj1020.1 . . . . 5 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
3 bnj1020.2 . . . . 5 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
4 bnj1020.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
5 bnj1020.4 . . . . 5 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
6 bnj1020.5 . . . . 5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
7 bnj1020.6 . . . . 5 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
8 bnj1020.7 . . . . 5 (𝜑′[𝑝 / 𝑛]𝜑)
9 bnj1020.8 . . . . 5 (𝜓′[𝑝 / 𝑛]𝜓)
10 bnj1020.9 . . . . 5 (𝜒′[𝑝 / 𝑛]𝜒)
11 bnj1020.10 . . . . 5 (𝜑″[𝐺 / 𝑓]𝜑′)
12 bnj1020.11 . . . . 5 (𝜓″[𝐺 / 𝑓]𝜓′)
13 bnj1020.12 . . . . 5 (𝜒″[𝐺 / 𝑓]𝜒′)
14 bnj1020.13 . . . . 5 𝐷 = (ω ∖ {∅})
15 bnj1020.15 . . . . 5 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
16 bnj1020.16 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
17 bnj1020.14 . . . . . . 7 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
182, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 15, 16bnj998 32256 . . . . . 6 ((𝜃𝜒𝜏𝜂) → 𝜒″)
194, 6, 7, 14, 18bnj1001 32258 . . . . 5 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
202, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19bnj1006 32259 . . . 4 ((𝜃𝜒𝜏𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
2120exlimiv 1932 . . 3 (∃𝑝(𝜃𝜒𝜏𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
221, 21sylbir 238 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
23 bnj1020.26 . . 3 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
242, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 15, 16, 23, 18, 19bnj1018 32263 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
2522, 24sstrd 3963 1 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wral 3133  wrex 3134  [wsbc 3758  cdif 3916  cun 3917  wss 3919  c0 4276  {csn 4550  cop 4556   ciun 4906  suc csuc 6181   Fn wfn 6339  cfv 6344  ωcom 7571  w-bnj17 31983   predc-bnj14 31985   FrSe w-bnj15 31989   trClc-bnj18 31991
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7452  ax-reg 9049
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-om 7572  df-bnj17 31984  df-bnj14 31986  df-bnj13 31988  df-bnj15 31990  df-bnj18 31992
This theorem is referenced by:  bnj907  32266
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