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Theorem bnj1021 33635
Description: Technical lemma for bnj69 33679. 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 33625 . . 3 𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂))
10 anclb 547 . . . . . 6 ((𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃 ∧ (𝜒𝜏𝜂))))
11 bnj252 33372 . . . . . . 7 ((𝜃𝜒𝜏𝜂) ↔ (𝜃 ∧ (𝜒𝜏𝜂)))
1211imbi2i 336 . . . . . 6 ((𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃 ∧ (𝜒𝜏𝜂))))
1310, 12bitr4i 278 . . . . 5 ((𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃𝜒𝜏𝜂)))
14132exbii 1852 . . . 4 (∃𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)))
15143exbii 1853 . . 3 (∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)))
169, 15mpbi 229 . 2 𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂))
17 19.37v 1996 . . . . 5 (∃𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → ∃𝑝(𝜃𝜒𝜏𝜂)))
18 bnj1019 33448 . . . . . 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 1852 . . 3 (∃𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ ∃𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
22212exbii 1852 . 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 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3061  wrex 3070  cdif 3908  c0 4283  {csn 4587   ciun 4955  suc csuc 6320   Fn wfn 6492  cfv 6497  ωcom 7803  w-bnj17 33355   predc-bnj14 33357   FrSe w-bnj15 33361   trClc-bnj18 33363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-tr 5224  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-fn 6500  df-om 7804  df-bnj17 33356  df-bnj18 33364
This theorem is referenced by:  bnj907  33636
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