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Theorem bnj1052 31342
 Description: Technical lemma for bnj69 31377. 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
bnj1052.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1052.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1052.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1052.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))
bnj1052.5 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1052.6 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
bnj1052.7 𝐷 = (ω ∖ {∅})
bnj1052.8 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1052.9 (𝜂 ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))
bnj1052.10 (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
bnj1052.37 ((𝜃𝜏𝜒𝜁) → ( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)))
Assertion
Ref Expression
bnj1052 ((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 bnj1052
StepHypRef Expression
1 bnj1052.1 . 2 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj1052.2 . 2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj1052.3 . 2 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj1052.4 . 2 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))
5 bnj1052.5 . 2 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
6 bnj1052.6 . 2 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
7 bnj1052.7 . 2 𝐷 = (ω ∖ {∅})
8 bnj1052.8 . 2 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
9 19.23vv 2013 . . . . 5 (∀𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ (∃𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
109albii 1888 . . . 4 (∀𝑓𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ ∀𝑓(∃𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
11 19.23v 2012 . . . 4 (∀𝑓(∃𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
1210, 11bitri 264 . . 3 (∀𝑓𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
13 bnj1052.37 . . . . 5 ((𝜃𝜏𝜒𝜁) → ( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)))
14 vex 3335 . . . . . . . . 9 𝑛 ∈ V
15 bnj1052.10 . . . . . . . . 9 (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
1614, 15bnj110 31227 . . . . . . . 8 (( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)) → ∀𝑖𝑛 𝜂)
17 bnj1052.9 . . . . . . . . 9 (𝜂 ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))
186, 17bnj1049 31341 . . . . . . . 8 (∀𝑖𝑛 𝜂 ↔ ∀𝑖𝜂)
1916, 18sylib 208 . . . . . . 7 (( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)) → ∀𝑖𝜂)
201919.21bi 2198 . . . . . 6 (( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)) → 𝜂)
2120, 17sylib 208 . . . . 5 (( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)) → ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))
2213, 21mpcom 38 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
2322gen2 1864 . . 3 𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
2412, 23mpgbi 1866 . 2 (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵)
251, 2, 3, 4, 5, 6, 7, 8, 24bnj1034 31337 1 ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072  ∀wal 1622   = wceq 1624  ∃wex 1845   ∈ wcel 2131  {cab 2738  ∀wral 3042  ∃wrex 3043  Vcvv 3332  [wsbc 3568   ∖ cdif 3704   ⊆ wss 3707  ∅c0 4050  {csn 4313  ∪ ciun 4664   class class class wbr 4796   E cep 5170   Fr wfr 5214  suc csuc 5878   Fn wfn 6036  ‘cfv 6041  ωcom 7222   ∧ w-bnj17 31053   predc-bnj14 31055   FrSe w-bnj15 31059   trClc-bnj18 31061   TrFow-bnj19 31063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-iun 4666  df-br 4797  df-fr 5217  df-fn 6044  df-bnj17 31054  df-bnj18 31062 This theorem is referenced by:  bnj1053  31343
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