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Theorem bnj1021 30777
Description: Technical lemma for bnj69 30821. 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 30768 . . 3 𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂))
10 anclb 569 . . . . . 6 ((𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃 ∧ (𝜒𝜏𝜂))))
11 bnj252 30511 . . . . . . 7 ((𝜃𝜒𝜏𝜂) ↔ (𝜃 ∧ (𝜒𝜏𝜂)))
1211imbi2i 326 . . . . . 6 ((𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃 ∧ (𝜒𝜏𝜂))))
1310, 12bitr4i 267 . . . . 5 ((𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃𝜒𝜏𝜂)))
14132exbii 1772 . . . 4 (∃𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)))
15143exbii 1773 . . 3 (∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)))
169, 15mpbi 220 . 2 𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂))
17 19.37v 1907 . . . . 5 (∃𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → ∃𝑝(𝜃𝜒𝜏𝜂)))
18 bnj1019 30593 . . . . . 6 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
1918imbi2i 326 . . . . 5 ((𝜃 → ∃𝑝(𝜃𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
2017, 19bitri 264 . . . 4 (∃𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
21202exbii 1772 . . 3 (∃𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ ∃𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
22212exbii 1772 . 2 (∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ ∃𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
2316, 22mpbi 220 1 𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  cdif 3556  c0 3896  {csn 4153   ciun 4490  suc csuc 5689   Fn wfn 5847  cfv 5852  ωcom 7019  w-bnj17 30494   predc-bnj14 30496   FrSe w-bnj15 30500   trClc-bnj18 30502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-fn 5855  df-om 7020  df-bnj17 30495  df-bnj18 30503
This theorem is referenced by:  bnj907  30778
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