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Theorem bnj1030 31572
Description: Technical lemma for bnj69 31595. 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
bnj1030.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1030.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1030.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1030.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))
bnj1030.5 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1030.6 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
bnj1030.7 𝐷 = (ω ∖ {∅})
bnj1030.8 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1030.9 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
bnj1030.10 (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
bnj1030.11 (𝜑′[𝑗 / 𝑖]𝜑)
bnj1030.12 (𝜓′[𝑗 / 𝑖]𝜓)
bnj1030.13 (𝜒′[𝑗 / 𝑖]𝜒)
bnj1030.14 (𝜃′[𝑗 / 𝑖]𝜃)
bnj1030.15 (𝜏′[𝑗 / 𝑖]𝜏)
bnj1030.16 (𝜁′[𝑗 / 𝑖]𝜁)
bnj1030.17 (𝜂′[𝑗 / 𝑖]𝜂)
bnj1030.18 (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))
bnj1030.19 (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))
Assertion
Ref Expression
bnj1030 ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑗,𝑛,𝑦   𝑧,𝐴,𝑓,𝑖,𝑛   𝐵,𝑓,𝑖,𝑛,𝑦   𝑧,𝐵   𝐷,𝑖,𝑗   𝑅,𝑓,𝑖,𝑗,𝑛,𝑦   𝑧,𝑅   𝑓,𝑋,𝑖,𝑛,𝑦   𝑧,𝑋   𝜒,𝑗   𝜂,𝑗   𝜏,𝑓,𝑖,𝑗,𝑛   𝜃,𝑓,𝑖,𝑗,𝑛   𝜑,𝑖   𝜏,𝑧   𝜃,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑗,𝑛)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑛)   𝜃(𝑦)   𝜏(𝑦)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑛)   𝜁(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜎(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜌(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝐵(𝑗)   𝐷(𝑦,𝑧,𝑓,𝑛)   𝐾(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝑋(𝑗)   𝜑′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜓′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜒′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜃′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜏′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜂′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜁′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜑0(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1030
StepHypRef Expression
1 bnj1030.1 . 2 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj1030.2 . 2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj1030.3 . 2 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj1030.4 . 2 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))
5 bnj1030.5 . 2 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
6 bnj1030.6 . 2 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
7 bnj1030.7 . 2 𝐷 = (ω ∖ {∅})
8 bnj1030.8 . 2 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
9 19.23vv 2039 . . . . 5 (∀𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ (∃𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
109albii 1915 . . . 4 (∀𝑓𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ ∀𝑓(∃𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
11 19.23v 2038 . . . 4 (∀𝑓(∃𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
1210, 11bitri 267 . . 3 (∀𝑓𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
13 bnj1030.9 . . . . 5 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
147bnj1071 31562 . . . . . . . 8 (𝑛𝐷 → E Fr 𝑛)
153, 14bnj769 31349 . . . . . . 7 (𝜒 → E Fr 𝑛)
1615bnj707 31342 . . . . . 6 ((𝜃𝜏𝜒𝜁) → E Fr 𝑛)
17 bnj1030.10 . . . . . . 7 (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
18 bnj1030.17 . . . . . . 7 (𝜂′[𝑗 / 𝑖]𝜂)
19 bnj1030.18 . . . . . . 7 (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))
20 bnj1030.19 . . . . . . 7 (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))
212, 8, 13, 18bnj1123 31571 . . . . . . . . . 10 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
222, 3, 5, 7, 19, 20, 21bnj1118 31569 . . . . . . . . 9 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
231, 3, 5bnj1097 31566 . . . . . . . . 9 ((𝑖 = ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
2422, 23bnj1109 31374 . . . . . . . 8 𝑗(((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑓𝑖) ⊆ 𝐵)
2524, 2, 3bnj1093 31565 . . . . . . 7 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))
2613, 17, 18, 19, 20, 25bnj1090 31564 . . . . . 6 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 (𝜌𝜂))
27 vex 3388 . . . . . . 7 𝑛 ∈ V
2827, 17bnj110 31445 . . . . . 6 (( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)) → ∀𝑖𝑛 𝜂)
2916, 26, 28syl2anc 580 . . . . 5 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 𝜂)
304, 5, 3, 6, 13, 29, 8bnj1121 31570 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
3130gen2 1892 . . 3 𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
3212, 31mpgbi 1894 . 2 (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵)
331, 2, 3, 4, 5, 6, 7, 8, 32bnj1034 31555 1 ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108  wal 1651   = wceq 1653  wex 1875  wcel 2157  {cab 2785  wral 3089  wrex 3090  Vcvv 3385  [wsbc 3633  cdif 3766  wss 3769  c0 4115  {csn 4368   ciun 4710   class class class wbr 4843   E cep 5224   Fr wfr 5268  dom cdm 5312  suc csuc 5943   Fn wfn 6096  cfv 6101  ωcom 7299  w-bnj17 31272   predc-bnj14 31274   FrSe w-bnj15 31278   trClc-bnj18 31280   TrFow-bnj19 31282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-tr 4946  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fn 6104  df-fv 6109  df-om 7300  df-bnj17 31273  df-bnj18 31281  df-bnj19 31283
This theorem is referenced by:  bnj1124  31573
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