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Theorem bnj1053 30779
Description: Technical lemma for bnj69 30813. 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
bnj1053.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1053.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1053.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1053.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))
bnj1053.5 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1053.6 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
bnj1053.7 𝐷 = (ω ∖ {∅})
bnj1053.8 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1053.9 (𝜂 ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))
bnj1053.10 (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
bnj1053.37 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 (𝜌𝜂))
Assertion
Ref Expression
bnj1053 ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝑧,𝐴,𝑓,𝑖,𝑛   𝐵,𝑓,𝑖,𝑛,𝑧   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑧,𝑅   𝑓,𝑋,𝑖,𝑛,𝑦   𝑧,𝑋   𝜂,𝑗   𝜏,𝑓,𝑖,𝑛,𝑧   𝜃,𝑓,𝑖,𝑛,𝑧   𝑖,𝑗,𝑛   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑗,𝑛)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜃(𝑦,𝑗)   𝜏(𝑦,𝑗)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑛)   𝜁(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜌(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝐴(𝑗)   𝐵(𝑦,𝑗)   𝐷(𝑦,𝑧,𝑓,𝑗,𝑛)   𝑅(𝑗)   𝐾(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝑋(𝑗)

Proof of Theorem bnj1053
StepHypRef Expression
1 bnj1053.1 . 2 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj1053.2 . 2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj1053.3 . 2 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj1053.4 . 2 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))
5 bnj1053.5 . 2 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
6 bnj1053.6 . 2 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
7 bnj1053.7 . 2 𝐷 = (ω ∖ {∅})
8 bnj1053.8 . 2 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
9 bnj1053.9 . 2 (𝜂 ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))
10 bnj1053.10 . 2 (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
117bnj923 30573 . . . . . 6 (𝑛𝐷𝑛 ∈ ω)
12 nnord 7027 . . . . . 6 (𝑛 ∈ ω → Ord 𝑛)
13 ordfr 5702 . . . . . 6 (Ord 𝑛 → E Fr 𝑛)
1411, 12, 133syl 18 . . . . 5 (𝑛𝐷 → E Fr 𝑛)
153, 14bnj769 30567 . . . 4 (𝜒 → E Fr 𝑛)
1615bnj707 30560 . . 3 ((𝜃𝜏𝜒𝜁) → E Fr 𝑛)
17 bnj1053.37 . . 3 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 (𝜌𝜂))
1816, 17jca 554 . 2 ((𝜃𝜏𝜒𝜁) → ( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)))
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18bnj1052 30778 1 ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3189  [wsbc 3421  cdif 3556  wss 3559  c0 3896  {csn 4153   ciun 4490   class class class wbr 4618   E cep 4988   Fr wfr 5035  Ord word 5686  suc csuc 5689   Fn wfn 5847  cfv 5852  ωcom 7019  w-bnj17 30486   predc-bnj14 30488   FrSe w-bnj15 30492   trClc-bnj18 30494   TrFow-bnj19 30496
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-fal 1486  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-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  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 30487  df-bnj18 30495
This theorem is referenced by: (None)
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