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Theorem bnj1053 31590
Description: Technical lemma for bnj69 31624. 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 31384 . . . . . 6 (𝑛𝐷𝑛 ∈ ω)
12 nnord 7334 . . . . . 6 (𝑛 ∈ ω → Ord 𝑛)
13 ordfr 5978 . . . . . 6 (Ord 𝑛 → E Fr 𝑛)
1411, 12, 133syl 18 . . . . 5 (𝑛𝐷 → E Fr 𝑛)
153, 14bnj769 31378 . . . 4 (𝜒 → E Fr 𝑛)
1615bnj707 31371 . . 3 ((𝜃𝜏𝜒𝜁) → E Fr 𝑛)
17 bnj1053.37 . . 3 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 (𝜌𝜂))
1816, 17jca 509 . 2 ((𝜃𝜏𝜒𝜁) → ( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)))
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18bnj1052 31589 1 ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  {cab 2811  wral 3117  wrex 3118  Vcvv 3414  [wsbc 3662  cdif 3795  wss 3798  c0 4144  {csn 4397   ciun 4740   class class class wbr 4873   E cep 5254   Fr wfr 5298  Ord word 5962  suc csuc 5965   Fn wfn 6118  cfv 6123  ωcom 7326  w-bnj17 31301   predc-bnj14 31303   FrSe w-bnj15 31307   trClc-bnj18 31309   TrFow-bnj19 31311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-tr 4976  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-fn 6126  df-om 7327  df-bnj17 31302  df-bnj18 31310
This theorem is referenced by: (None)
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