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Theorem bnj1006 32225
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
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 499 . . . 4 (𝜂𝑦 ∈ (𝑓𝑖))
32bnj708 32020 . . 3 ((𝜃𝜒𝜏𝜂) → 𝑦 ∈ (𝑓𝑖))
4 bnj1006.4 . . . . . . . 8 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
5 bnj253 31967 . . . . . . . . 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 1140 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → (𝑅 FrSe 𝐴𝑋𝐴))
74, 6sylbi 219 . . . . . . 7 (𝜃 → (𝑅 FrSe 𝐴𝑋𝐴))
87bnj705 32017 . . . . . 6 ((𝜃𝜒𝜏𝜂) → (𝑅 FrSe 𝐴𝑋𝐴))
9 bnj643 32013 . . . . . . 7 ((𝜃𝜒𝜏𝜂) → 𝜒)
10 bnj1006.5 . . . . . . . . 9 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
11 3simpc 1145 . . . . . . . . 9 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
1210, 11sylbi 219 . . . . . . . 8 (𝜏 → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
1312bnj707 32019 . . . . . . 7 ((𝜃𝜒𝜏𝜂) → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
14 3anass 1090 . . . . . . 7 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ↔ (𝜒 ∧ (𝑛 = suc 𝑚𝑝 = suc 𝑛)))
159, 13, 14sylanbrc 585 . . . . . 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 263 . . . . . . 7 ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓))
22 biid 263 . . . . . . 7 ((𝑛𝐷𝑝 = suc 𝑛𝑚𝑛) ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
2316, 17, 18, 19, 20, 21, 22bnj969 32211 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
248, 15, 23syl2anc 586 . . . . 5 ((𝜃𝜒𝜏𝜂) → 𝐶 ∈ V)
2518bnj1235 32069 . . . . . 6 (𝜒𝑓 Fn 𝑛)
2625bnj706 32018 . . . . 5 ((𝜃𝜒𝜏𝜂) → 𝑓 Fn 𝑛)
2710simp3bi 1142 . . . . . 6 (𝜏𝑝 = suc 𝑛)
2827bnj707 32019 . . . . 5 ((𝜃𝜒𝜏𝜂) → 𝑝 = suc 𝑛)
291simplbi 500 . . . . . 6 (𝜂𝑖𝑛)
3029bnj708 32020 . . . . 5 ((𝜃𝜒𝜏𝜂) → 𝑖𝑛)
3124, 26, 28, 30bnj951 32040 . . . 4 ((𝜃𝜒𝜏𝜂) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛))
32 bnj1006.16 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
3332bnj945 32038 . . . 4 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛) → (𝐺𝑖) = (𝑓𝑖))
3431, 33syl 17 . . 3 ((𝜃𝜒𝜏𝜂) → (𝐺𝑖) = (𝑓𝑖))
353, 34eleqtrrd 2914 . 2 ((𝜃𝜒𝜏𝜂) → 𝑦 ∈ (𝐺𝑖))
36 bnj1006.28 . . . . 5 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
3736anim1i 616 . . . 4 (((𝜃𝜒𝜏𝜂) ∧ 𝑦 ∈ (𝐺𝑖)) → ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑦 ∈ (𝐺𝑖)))
38 df-bnj17 31950 . . . 4 ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) ↔ ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑦 ∈ (𝐺𝑖)))
3937, 38sylibr 236 . . 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 32223 . . 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 685 1 ((𝜃𝜒𝜏𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wral 3136  Vcvv 3493  [wsbc 3770  cdif 3931  cun 3932  wss 3934  c0 4289  {csn 4559  cop 4565   ciun 4910  suc csuc 6186   Fn wfn 6343  cfv 6348  ωcom 7572  w-bnj17 31949   predc-bnj14 31951   FrSe w-bnj15 31955   trClc-bnj18 31957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453  ax-reg 9048
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  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-reu 3143  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-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 7573  df-bnj17 31950  df-bnj14 31952  df-bnj13 31954  df-bnj15 31956
This theorem is referenced by:  bnj1020  32230
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