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Theorem bnj907 32136
Description: Technical lemma for bnj69 32179. 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
bnj907.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj907.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj907.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj907.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj907.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj907.6 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
bnj907.7 (𝜑′[𝑝 / 𝑛]𝜑)
bnj907.8 (𝜓′[𝑝 / 𝑛]𝜓)
bnj907.9 (𝜒′[𝑝 / 𝑛]𝜒)
bnj907.10 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj907.11 (𝜓″[𝐺 / 𝑓]𝜓′)
bnj907.12 (𝜒″[𝐺 / 𝑓]𝜒′)
bnj907.13 𝐷 = (ω ∖ {∅})
bnj907.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj907.15 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj907.16 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
Assertion
Ref Expression
bnj907 ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑝,𝑦   𝑧,𝐴,𝑦   𝐷,𝑓,𝑖,𝑛   𝑖,𝐺,𝑝   𝑅,𝑓,𝑖,𝑚,𝑛,𝑝,𝑦   𝑧,𝑅   𝑓,𝑋,𝑖,𝑚,𝑛,𝑦   𝑧,𝑋   𝜒,𝑚,𝑝   𝜂,𝑚,𝑝   𝜃,𝑓,𝑖,𝑚,𝑛,𝑝   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑛)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑛)   𝐵(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑚,𝑝)   𝐺(𝑦,𝑧,𝑓,𝑚,𝑛)   𝑋(𝑝)   𝜑′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj907
StepHypRef Expression
1 bnj907.4 . 2 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
2 bnj907.1 . . . . . . . . 9 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
3 bnj907.2 . . . . . . . . 9 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
4 bnj907.3 . . . . . . . . 9 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
5 bnj907.5 . . . . . . . . 9 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
6 bnj907.6 . . . . . . . . 9 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
7 bnj907.13 . . . . . . . . 9 𝐷 = (ω ∖ {∅})
8 bnj907.14 . . . . . . . . 9 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
92, 3, 4, 1, 5, 6, 7, 8bnj1021 32135 . . . . . . . 8 𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
10 bnj907.7 . . . . . . . . . . . 12 (𝜑′[𝑝 / 𝑛]𝜑)
11 bnj907.8 . . . . . . . . . . . 12 (𝜓′[𝑝 / 𝑛]𝜓)
12 bnj907.9 . . . . . . . . . . . 12 (𝜒′[𝑝 / 𝑛]𝜒)
13 bnj907.10 . . . . . . . . . . . 12 (𝜑″[𝐺 / 𝑓]𝜑′)
14 bnj907.11 . . . . . . . . . . . 12 (𝜓″[𝐺 / 𝑓]𝜓′)
15 bnj907.12 . . . . . . . . . . . 12 (𝜒″[𝐺 / 𝑓]𝜒′)
16 bnj907.15 . . . . . . . . . . . 12 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
17 bnj907.16 . . . . . . . . . . . 12 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
18 vex 3495 . . . . . . . . . . . . . 14 𝑝 ∈ V
194, 10, 11, 12, 18bnj919 31937 . . . . . . . . . . . . 13 (𝜒′ ↔ (𝑝𝐷𝑓 Fn 𝑝𝜑′𝜓′))
2017bnj918 31936 . . . . . . . . . . . . 13 𝐺 ∈ V
2119, 13, 14, 15, 20bnj976 31948 . . . . . . . . . . . 12 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
222, 3, 4, 1, 5, 6, 10, 11, 12, 13, 14, 15, 7, 8, 16, 17, 21bnj1020 32134 . . . . . . . . . . 11 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
2322ax-gen 1787 . . . . . . . . . 10 𝑚((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
24 19.29r 1866 . . . . . . . . . . 11 ((∃𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)) ∧ ∀𝑚((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))) → ∃𝑚((𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)) ∧ ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))))
25 pm3.33 761 . . . . . . . . . . 11 (((𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)) ∧ ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))) → (𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
2624, 25bnj593 31915 . . . . . . . . . 10 ((∃𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)) ∧ ∀𝑚((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))) → ∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
2723, 26mpan2 687 . . . . . . . . 9 (∃𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)) → ∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
28272eximi 1827 . . . . . . . 8 (∃𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)) → ∃𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
299, 28bnj101 31892 . . . . . . 7 𝑓𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
30 19.9v 1979 . . . . . . 7 (∃𝑓𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ∃𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
3129, 30mpbi 231 . . . . . 6 𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
32 19.9v 1979 . . . . . 6 (∃𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ∃𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
3331, 32mpbi 231 . . . . 5 𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
34 19.9v 1979 . . . . 5 (∃𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
3533, 34mpbi 231 . . . 4 𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
36 19.9v 1979 . . . 4 (∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ (𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
3735, 36mpbi 231 . . 3 (𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
381bnj1254 31980 . . 3 (𝜃𝑧 ∈ pred(𝑦, 𝐴, 𝑅))
3937, 38sseldd 3965 . 2 (𝜃𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
401, 39bnj978 32120 1 ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079  wal 1526   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wral 3135  wrex 3136  [wsbc 3769  cdif 3930  cun 3931  wss 3933  c0 4288  {csn 4557  cop 4563   ciun 4910  suc csuc 6186   Fn wfn 6343  cfv 6348  ωcom 7569  w-bnj17 31855   predc-bnj14 31857   FrSe w-bnj15 31861   trClc-bnj18 31863   TrFow-bnj19 31865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450  ax-reg 9044
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-bnj17 31856  df-bnj14 31858  df-bnj13 31860  df-bnj15 31862  df-bnj18 31864  df-bnj19 31866
This theorem is referenced by:  bnj1029  32137
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