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Theorem bnj907 31334
 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
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 31333 . . . . . . . 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 3335 . . . . . . . . . . . . . 14 𝑝 ∈ V
194, 10, 11, 12, 18bnj919 31136 . . . . . . . . . . . . 13 (𝜒′ ↔ (𝑝𝐷𝑓 Fn 𝑝𝜑′𝜓′))
2017bnj918 31135 . . . . . . . . . . . . 13 𝐺 ∈ V
2119, 13, 14, 15, 20bnj976 31147 . . . . . . . . . . . 12 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
222, 3, 4, 1, 5, 6, 10, 11, 12, 13, 14, 15, 7, 8, 16, 17, 21bnj1020 31332 . . . . . . . . . . 11 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
2322ax-gen 1863 . . . . . . . . . 10 𝑚((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
24 19.29r 1943 . . . . . . . . . . 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 pm3.33 610 . . . . . . . . . . 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 31114 . . . . . . . . . 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 709 . . . . . . . . 9 (∃𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)) → ∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
28272eximi 1904 . . . . . . . 8 (∃𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)) → ∃𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
299, 28bnj101 31090 . . . . . . 7 𝑓𝑛𝑖𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
30 19.9v 2054 . . . . . . 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 2054 . . . . . 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 2054 . . . . 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 2054 . . . 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 31179 . . 3 (𝜃𝑧 ∈ pred(𝑦, 𝐴, 𝑅))
3937, 38sseldd 3737 . 2 (𝜃𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
401, 39bnj978 31318 1 ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))
 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  [wsbc 3568   ∖ cdif 3704   ∪ cun 3705   ⊆ wss 3707  ∅c0 4050  {csn 4313  ⟨cop 4319  ∪ ciun 4664  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-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pr 5047  ax-un 7106  ax-reg 8654 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  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-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-om 7223  df-bnj17 31054  df-bnj14 31056  df-bnj13 31058  df-bnj15 31060  df-bnj18 31062  df-bnj19 31064 This theorem is referenced by:  bnj1029  31335
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