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Theorem bnj1053 32248
 Description: Technical lemma for bnj69 32282. 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 32039 . . . . . 6 (𝑛𝐷𝑛 ∈ ω)
12 nnord 7587 . . . . . 6 (𝑛 ∈ ω → Ord 𝑛)
13 ordfr 6205 . . . . . 6 (Ord 𝑛 → E Fr 𝑛)
1411, 12, 133syl 18 . . . . 5 (𝑛𝐷 → E Fr 𝑛)
153, 14bnj769 32033 . . . 4 (𝜒 → E Fr 𝑛)
1615bnj707 32026 . . 3 ((𝜃𝜏𝜒𝜁) → E Fr 𝑛)
17 bnj1053.37 . . 3 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 (𝜌𝜂))
1816, 17jca 514 . 2 ((𝜃𝜏𝜒𝜁) → ( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)))
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18bnj1052 32247 1 ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  ∃wrex 3139  Vcvv 3494  [wsbc 3771   ∖ cdif 3932   ⊆ wss 3935  ∅c0 4290  {csn 4566  ∪ ciun 4918   class class class wbr 5065   E cep 5463   Fr wfr 5510  Ord word 6189  suc csuc 6192   Fn wfn 6349  ‘cfv 6354  ωcom 7579   ∧ w-bnj17 31956   predc-bnj14 31958   FrSe w-bnj15 31962   trClc-bnj18 31964   TrFow-bnj19 31966 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329  ax-un 7460 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-tr 5172  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-fn 6357  df-om 7580  df-bnj17 31957  df-bnj18 31965 This theorem is referenced by: (None)
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