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Theorem bnj1006 34952
Description: Technical lemma for bnj69 35002. 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
bnj1006.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1006.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1006.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1006.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj1006.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj1006.6 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
bnj1006.7 (𝜑′[𝑝 / 𝑛]𝜑)
bnj1006.8 (𝜓′[𝑝 / 𝑛]𝜓)
bnj1006.9 (𝜒′[𝑝 / 𝑛]𝜒)
bnj1006.10 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj1006.11 (𝜓″[𝐺 / 𝑓]𝜓′)
bnj1006.12 (𝜒″[𝐺 / 𝑓]𝜒′)
bnj1006.13 𝐷 = (ω ∖ {∅})
bnj1006.15 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj1006.16 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj1006.28 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
Assertion
Ref Expression
bnj1006 ((𝜃𝜒𝜏𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐷,𝑓,𝑛   𝑖,𝐺   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑓,𝑋,𝑛   𝑓,𝑝,𝑖,𝑛
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜃(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜏(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑧,𝑝)   𝐶(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑖,𝑚,𝑝)   𝑅(𝑧,𝑝)   𝐺(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑧,𝑖,𝑚,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 bnj1006
StepHypRef Expression
1 bnj1006.6 . . . . 5 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
21simprbi 496 . . . 4 (𝜂𝑦 ∈ (𝑓𝑖))
32bnj708 34748 . . 3 ((𝜃𝜒𝜏𝜂) → 𝑦 ∈ (𝑓𝑖))
4 bnj1006.4 . . . . . . . 8 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
5 bnj253 34696 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
65simp1bi 1144 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → (𝑅 FrSe 𝐴𝑋𝐴))
74, 6sylbi 217 . . . . . . 7 (𝜃 → (𝑅 FrSe 𝐴𝑋𝐴))
87bnj705 34745 . . . . . 6 ((𝜃𝜒𝜏𝜂) → (𝑅 FrSe 𝐴𝑋𝐴))
9 bnj643 34741 . . . . . . 7 ((𝜃𝜒𝜏𝜂) → 𝜒)
10 bnj1006.5 . . . . . . . . 9 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
11 3simpc 1149 . . . . . . . . 9 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
1210, 11sylbi 217 . . . . . . . 8 (𝜏 → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
1312bnj707 34747 . . . . . . 7 ((𝜃𝜒𝜏𝜂) → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
14 3anass 1094 . . . . . . 7 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ↔ (𝜒 ∧ (𝑛 = suc 𝑚𝑝 = suc 𝑛)))
159, 13, 14sylanbrc 583 . . . . . 6 ((𝜃𝜒𝜏𝜂) → (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛))
16 bnj1006.1 . . . . . . 7 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
17 bnj1006.2 . . . . . . 7 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
18 bnj1006.3 . . . . . . 7 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
19 bnj1006.13 . . . . . . 7 𝐷 = (ω ∖ {∅})
20 bnj1006.15 . . . . . . 7 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
21 biid 261 . . . . . . 7 ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓))
22 biid 261 . . . . . . 7 ((𝑛𝐷𝑝 = suc 𝑛𝑚𝑛) ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
2316, 17, 18, 19, 20, 21, 22bnj969 34938 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
248, 15, 23syl2anc 584 . . . . 5 ((𝜃𝜒𝜏𝜂) → 𝐶 ∈ V)
2518bnj1235 34796 . . . . . 6 (𝜒𝑓 Fn 𝑛)
2625bnj706 34746 . . . . 5 ((𝜃𝜒𝜏𝜂) → 𝑓 Fn 𝑛)
2710simp3bi 1146 . . . . . 6 (𝜏𝑝 = suc 𝑛)
2827bnj707 34747 . . . . 5 ((𝜃𝜒𝜏𝜂) → 𝑝 = suc 𝑛)
291simplbi 497 . . . . . 6 (𝜂𝑖𝑛)
3029bnj708 34748 . . . . 5 ((𝜃𝜒𝜏𝜂) → 𝑖𝑛)
3124, 26, 28, 30bnj951 34767 . . . 4 ((𝜃𝜒𝜏𝜂) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛))
32 bnj1006.16 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
3332bnj945 34765 . . . 4 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛) → (𝐺𝑖) = (𝑓𝑖))
3431, 33syl 17 . . 3 ((𝜃𝜒𝜏𝜂) → (𝐺𝑖) = (𝑓𝑖))
353, 34eleqtrrd 2841 . 2 ((𝜃𝜒𝜏𝜂) → 𝑦 ∈ (𝐺𝑖))
36 bnj1006.28 . . . . 5 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
3736anim1i 615 . . . 4 (((𝜃𝜒𝜏𝜂) ∧ 𝑦 ∈ (𝐺𝑖)) → ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑦 ∈ (𝐺𝑖)))
38 df-bnj17 34679 . . . 4 ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) ↔ ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑦 ∈ (𝐺𝑖)))
3937, 38sylibr 234 . . 3 (((𝜃𝜒𝜏𝜂) ∧ 𝑦 ∈ (𝐺𝑖)) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)))
40 bnj1006.7 . . . 4 (𝜑′[𝑝 / 𝑛]𝜑)
41 bnj1006.8 . . . 4 (𝜓′[𝑝 / 𝑛]𝜓)
42 bnj1006.9 . . . 4 (𝜒′[𝑝 / 𝑛]𝜒)
43 bnj1006.10 . . . 4 (𝜑″[𝐺 / 𝑓]𝜑′)
44 bnj1006.11 . . . 4 (𝜓″[𝐺 / 𝑓]𝜓′)
45 bnj1006.12 . . . 4 (𝜒″[𝐺 / 𝑓]𝜒′)
4616, 17, 18, 40, 41, 42, 43, 44, 45, 20, 32bnj999 34950 . . 3 ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
4739, 46syl 17 . 2 (((𝜃𝜒𝜏𝜂) ∧ 𝑦 ∈ (𝐺𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
4835, 47mpdan 687 1 ((𝜃𝜒𝜏𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  [wsbc 3790  cdif 3959  cun 3960  wss 3962  c0 4338  {csn 4630  cop 4636   ciun 4995  suc csuc 6387   Fn wfn 6557  cfv 6562  ωcom 7886  w-bnj17 34678   predc-bnj14 34680   FrSe w-bnj15 34684   trClc-bnj18 34686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753  ax-reg 9629
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-res 5700  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570  df-om 7887  df-bnj17 34679  df-bnj14 34681  df-bnj13 34683  df-bnj15 34685
This theorem is referenced by:  bnj1020  34957
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