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Theorem bnj1021 34505
Description: Technical lemma for bnj69 34549. 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
bnj1021.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1021.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1021.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1021.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj1021.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj1021.6 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
bnj1021.13 𝐷 = (ω ∖ {∅})
bnj1021.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
Assertion
Ref Expression
bnj1021 𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
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 bnj1021
StepHypRef Expression
1 bnj1021.1 . . . 4 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj1021.2 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj1021.3 . . . 4 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj1021.4 . . . 4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
5 bnj1021.5 . . . 4 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
6 bnj1021.6 . . . 4 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
7 bnj1021.13 . . . 4 𝐷 = (ω ∖ {∅})
8 bnj1021.14 . . . 4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
91, 2, 3, 4, 5, 6, 7, 8bnj996 34495 . . 3 𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂))
10 anclb 545 . . . . . 6 ((𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃 ∧ (𝜒𝜏𝜂))))
11 bnj252 34242 . . . . . . 7 ((𝜃𝜒𝜏𝜂) ↔ (𝜃 ∧ (𝜒𝜏𝜂)))
1211imbi2i 336 . . . . . 6 ((𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃 ∧ (𝜒𝜏𝜂))))
1310, 12bitr4i 278 . . . . 5 ((𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃𝜒𝜏𝜂)))
14132exbii 1843 . . . 4 (∃𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)))
15143exbii 1844 . . 3 (∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)))
169, 15mpbi 229 . 2 𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂))
17 19.37v 1987 . . . . 5 (∃𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → ∃𝑝(𝜃𝜒𝜏𝜂)))
18 bnj1019 34318 . . . . . 6 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
1918imbi2i 336 . . . . 5 ((𝜃 → ∃𝑝(𝜃𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
2017, 19bitri 275 . . . 4 (∃𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
21202exbii 1843 . . 3 (∃𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ ∃𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
22212exbii 1843 . 2 (∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ ∃𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
2316, 22mpbi 229 1 𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2703  wral 3055  wrex 3064  cdif 3940  c0 4317  {csn 4623   ciun 4990  suc csuc 6359   Fn wfn 6531  cfv 6536  ωcom 7851  w-bnj17 34225   predc-bnj14 34227   FrSe w-bnj15 34231   trClc-bnj18 34233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-fn 6539  df-om 7852  df-bnj17 34226  df-bnj18 34234
This theorem is referenced by:  bnj907  34506
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