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Theorem bnj907 31020
Description: Technical lemma for bnj69 31063. 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 31019 . . . . . . . 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 3201 . . . . . . . . . . . . . 14 𝑝 ∈ V
194, 10, 11, 12, 18bnj919 30822 . . . . . . . . . . . . 13 (𝜒′ ↔ (𝑝𝐷𝑓 Fn 𝑝𝜑′𝜓′))
2017bnj918 30821 . . . . . . . . . . . . 13 𝐺 ∈ V
2119, 13, 14, 15, 20bnj976 30833 . . . . . . . . . . . 12 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
222, 3, 4, 1, 5, 6, 10, 11, 12, 13, 14, 15, 7, 8, 16, 17, 21bnj1020 31018 . . . . . . . . . . 11 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
2322ax-gen 1721 . . . . . . . . . 10 𝑚((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
24 19.29r 1801 . . . . . . . . . . 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 pm3.33 609 . . . . . . . . . . 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 30800 . . . . . . . . . 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 707 . . . . . . . . 9 (∃𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)) → ∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
28272eximi 1762 . . . . . . . 8 (∃𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)) → ∃𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
299, 28bnj101 30774 . . . . . . 7 𝑓𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
30 19.9v 1895 . . . . . . 7 (∃𝑓𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ∃𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
3129, 30mpbi 220 . . . . . 6 𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
32 19.9v 1895 . . . . . 6 (∃𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ∃𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
3331, 32mpbi 220 . . . . 5 𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
34 19.9v 1895 . . . . 5 (∃𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
3533, 34mpbi 220 . . . 4 𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
36 19.9v 1895 . . . 4 (∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ (𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
3735, 36mpbi 220 . . 3 (𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
381bnj1254 30865 . . 3 (𝜃𝑧 ∈ pred(𝑦, 𝐴, 𝑅))
3937, 38sseldd 3602 . 2 (𝜃𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
401, 39bnj978 31004 1 ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1037  wal 1480   = wceq 1482  wex 1703  wcel 1989  {cab 2607  wral 2911  wrex 2912  [wsbc 3433  cdif 3569  cun 3570  wss 3572  c0 3913  {csn 4175  cop 4181   ciun 4518  suc csuc 5723   Fn wfn 5881  cfv 5886  ωcom 7062  w-bnj17 30737   predc-bnj14 30739   FrSe w-bnj15 30743   trClc-bnj18 30745   TrFow-bnj19 30747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-un 6946  ax-reg 8494
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-om 7063  df-bnj17 30738  df-bnj14 30740  df-bnj13 30742  df-bnj15 30744  df-bnj18 30746  df-bnj19 30748
This theorem is referenced by:  bnj1029  31021
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