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Theorem bnj1030 32252
 Description: Technical lemma for bnj69 32275. 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 1937 . . . . 5 (∀𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ (∃𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
109albii 1813 . . . 4 (∀𝑓𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ ∀𝑓(∃𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
11 19.23v 1936 . . . 4 (∀𝑓(∃𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
1210, 11bitri 277 . . 3 (∀𝑓𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
13 bnj1030.9 . . . . 5 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
147bnj1071 32242 . . . . . . . 8 (𝑛𝐷 → E Fr 𝑛)
153, 14bnj769 32026 . . . . . . 7 (𝜒 → E Fr 𝑛)
1615bnj707 32019 . . . . . 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 32251 . . . . . . . . . 10 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
222, 3, 5, 7, 19, 20, 21bnj1118 32249 . . . . . . . . 9 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
231, 3, 5bnj1097 32246 . . . . . . . . 9 ((𝑖 = ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
2422, 23bnj1109 32051 . . . . . . . 8 𝑗(((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑓𝑖) ⊆ 𝐵)
2524, 2, 3bnj1093 32245 . . . . . . 7 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))
2613, 17, 18, 19, 20, 25bnj1090 32244 . . . . . 6 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 (𝜌𝜂))
27 vex 3496 . . . . . . 7 𝑛 ∈ V
2827, 17bnj110 32123 . . . . . 6 (( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)) → ∀𝑖𝑛 𝜂)
2916, 26, 28syl2anc 586 . . . . 5 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 𝜂)
304, 5, 3, 6, 13, 29, 8bnj1121 32250 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
3130gen2 1790 . . 3 𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
3212, 31mpgbi 1792 . 2 (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵)
331, 2, 3, 4, 5, 6, 7, 8, 32bnj1034 32235 1 ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1081  ∀wal 1528   = wceq 1530  ∃wex 1773   ∈ wcel 2107  {cab 2797  ∀wral 3136  ∃wrex 3137  Vcvv 3493  [wsbc 3770   ∖ cdif 3931   ⊆ wss 3934  ∅c0 4289  {csn 4559  ∪ ciun 4910   class class class wbr 5057   E cep 5457   Fr wfr 5504  dom cdm 5548  suc csuc 6186   Fn wfn 6343  ‘cfv 6348  ωcom 7572   ∧ w-bnj17 31949   predc-bnj14 31951   FrSe w-bnj15 31955   trClc-bnj18 31957   TrFow-bnj19 31959 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fn 6351  df-fv 6356  df-om 7573  df-bnj17 31950  df-bnj18 31958  df-bnj19 31960 This theorem is referenced by:  bnj1124  32253
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