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Theorem bnj996 32255
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
bnj996.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj996.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj996.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj996.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj996.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj996.6 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
bnj996.13 𝐷 = (ω ∖ {∅})
bnj996.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
Assertion
Ref Expression
bnj996 𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝜒,𝑚,𝑝   𝜂,𝑚,𝑝   𝜃,𝑓,𝑖,𝑛   𝜑,𝑖   𝑚,𝑛,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,𝑛)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑛)   𝐴(𝑧,𝑚,𝑝)   𝐵(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝑅(𝑧,𝑚,𝑝)   𝑋(𝑧,𝑚,𝑝)

Proof of Theorem bnj996
StepHypRef Expression
1 bnj996.4 . . . . 5 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
2 bnj996.1 . . . . . 6 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
3 bnj996.2 . . . . . 6 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
4 bnj996.13 . . . . . 6 𝐷 = (ω ∖ {∅})
5 bnj996.14 . . . . . 6 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
6 bnj996.3 . . . . . 6 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
72, 3, 4, 5, 6bnj917 32233 . . . . 5 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))
81, 7bnj771 32062 . . . 4 (𝜃 → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))
9 3anass 1092 . . . . . 6 ((𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ (𝜒 ∧ (𝑖𝑛𝑦 ∈ (𝑓𝑖))))
10 bnj996.6 . . . . . . 7 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
1110anbi2i 625 . . . . . 6 ((𝜒𝜂) ↔ (𝜒 ∧ (𝑖𝑛𝑦 ∈ (𝑓𝑖))))
129, 11bitr4i 281 . . . . 5 ((𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ (𝜒𝜂))
13123exbii 1851 . . . 4 (∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ ∃𝑓𝑛𝑖(𝜒𝜂))
148, 13sylib 221 . . 3 (𝜃 → ∃𝑓𝑛𝑖(𝜒𝜂))
15 bnj996.5 . . . . . . . . . 10 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
166, 4, 15bnj986 32254 . . . . . . . . 9 (𝜒 → ∃𝑚𝑝𝜏)
1716ancli 552 . . . . . . . 8 (𝜒 → (𝜒 ∧ ∃𝑚𝑝𝜏))
18 19.42vv 1959 . . . . . . . 8 (∃𝑚𝑝(𝜒𝜏) ↔ (𝜒 ∧ ∃𝑚𝑝𝜏))
1917, 18sylibr 237 . . . . . . 7 (𝜒 → ∃𝑚𝑝(𝜒𝜏))
2019anim1i 617 . . . . . 6 ((𝜒𝜂) → (∃𝑚𝑝(𝜒𝜏) ∧ 𝜂))
21 19.41vv 1952 . . . . . 6 (∃𝑚𝑝((𝜒𝜏) ∧ 𝜂) ↔ (∃𝑚𝑝(𝜒𝜏) ∧ 𝜂))
2220, 21sylibr 237 . . . . 5 ((𝜒𝜂) → ∃𝑚𝑝((𝜒𝜏) ∧ 𝜂))
23 df-3an 1086 . . . . . 6 ((𝜒𝜏𝜂) ↔ ((𝜒𝜏) ∧ 𝜂))
24232exbii 1850 . . . . 5 (∃𝑚𝑝(𝜒𝜏𝜂) ↔ ∃𝑚𝑝((𝜒𝜏) ∧ 𝜂))
2522, 24sylibr 237 . . . 4 ((𝜒𝜂) → ∃𝑚𝑝(𝜒𝜏𝜂))
26252eximi 1837 . . 3 (∃𝑛𝑖(𝜒𝜂) → ∃𝑛𝑖𝑚𝑝(𝜒𝜏𝜂))
2714, 26bnj593 32043 . 2 (𝜃 → ∃𝑓𝑛𝑖𝑚𝑝(𝜒𝜏𝜂))
28 19.37v 1999 . . . . . . . . . 10 (∃𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → ∃𝑝(𝜒𝜏𝜂)))
2928exbii 1849 . . . . . . . . 9 (∃𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑚(𝜃 → ∃𝑝(𝜒𝜏𝜂)))
3029bnj132 32023 . . . . . . . 8 (∃𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → ∃𝑚𝑝(𝜒𝜏𝜂)))
3130exbii 1849 . . . . . . 7 (∃𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑖(𝜃 → ∃𝑚𝑝(𝜒𝜏𝜂)))
3231bnj132 32023 . . . . . 6 (∃𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → ∃𝑖𝑚𝑝(𝜒𝜏𝜂)))
3332exbii 1849 . . . . 5 (∃𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑛(𝜃 → ∃𝑖𝑚𝑝(𝜒𝜏𝜂)))
3433bnj132 32023 . . . 4 (∃𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → ∃𝑛𝑖𝑚𝑝(𝜒𝜏𝜂)))
3534exbii 1849 . . 3 (∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑓(𝜃 → ∃𝑛𝑖𝑚𝑝(𝜒𝜏𝜂)))
3635bnj132 32023 . 2 (∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → ∃𝑓𝑛𝑖𝑚𝑝(𝜒𝜏𝜂)))
3727, 36mpbir 234 1 𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂))
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  cdif 3916  c0 4276  {csn 4550   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-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7452
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-rab 3142  df-v 3482  df-sbc 3759  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-tr 5160  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-fn 6347  df-om 7572  df-bnj17 31984  df-bnj18 31992
This theorem is referenced by:  bnj1021  32265
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