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Theorem bnj1006 30283
 Description: Technical lemma for bnj69 30332. 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 479 . . . 4 (𝜂𝑦 ∈ (𝑓𝑖))
32bnj708 30080 . . 3 ((𝜃𝜒𝜏𝜂) → 𝑦 ∈ (𝑓𝑖))
4 bnj1006.4 . . . . . . . 8 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
5 bnj253 30023 . . . . . . . . 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 1069 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → (𝑅 FrSe 𝐴𝑋𝐴))
74, 6sylbi 206 . . . . . . 7 (𝜃 → (𝑅 FrSe 𝐴𝑋𝐴))
87bnj705 30077 . . . . . 6 ((𝜃𝜒𝜏𝜂) → (𝑅 FrSe 𝐴𝑋𝐴))
9 bnj643 30073 . . . . . . 7 ((𝜃𝜒𝜏𝜂) → 𝜒)
10 bnj1006.5 . . . . . . . . 9 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
11 3simpc 1053 . . . . . . . . 9 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
1210, 11sylbi 206 . . . . . . . 8 (𝜏 → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
1312bnj707 30079 . . . . . . 7 ((𝜃𝜒𝜏𝜂) → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
14 3anass 1035 . . . . . . 7 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ↔ (𝜒 ∧ (𝑛 = suc 𝑚𝑝 = suc 𝑛)))
159, 13, 14sylanbrc 695 . . . . . 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 250 . . . . . . 7 ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓))
22 biid 250 . . . . . . 7 ((𝑛𝐷𝑝 = suc 𝑛𝑚𝑛) ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
2316, 17, 18, 19, 20, 21, 22bnj969 30270 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
248, 15, 23syl2anc 691 . . . . 5 ((𝜃𝜒𝜏𝜂) → 𝐶 ∈ V)
2518bnj1235 30129 . . . . . 6 (𝜒𝑓 Fn 𝑛)
2625bnj706 30078 . . . . 5 ((𝜃𝜒𝜏𝜂) → 𝑓 Fn 𝑛)
2710simp3bi 1071 . . . . . 6 (𝜏𝑝 = suc 𝑛)
2827bnj707 30079 . . . . 5 ((𝜃𝜒𝜏𝜂) → 𝑝 = suc 𝑛)
291simplbi 475 . . . . . 6 (𝜂𝑖𝑛)
3029bnj708 30080 . . . . 5 ((𝜃𝜒𝜏𝜂) → 𝑖𝑛)
3124, 26, 28, 30bnj951 30100 . . . 4 ((𝜃𝜒𝜏𝜂) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛))
32 bnj1006.16 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
3332bnj945 30098 . . . 4 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛) → (𝐺𝑖) = (𝑓𝑖))
3431, 33syl 17 . . 3 ((𝜃𝜒𝜏𝜂) → (𝐺𝑖) = (𝑓𝑖))
353, 34eleqtrrd 2691 . 2 ((𝜃𝜒𝜏𝜂) → 𝑦 ∈ (𝐺𝑖))
36 bnj1006.28 . . . . 5 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
3736anim1i 590 . . . 4 (((𝜃𝜒𝜏𝜂) ∧ 𝑦 ∈ (𝐺𝑖)) → ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑦 ∈ (𝐺𝑖)))
38 df-bnj17 30006 . . . 4 ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) ↔ ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑦 ∈ (𝐺𝑖)))
3937, 38sylibr 223 . . 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 30281 . . 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 699 1 ((𝜃𝜒𝜏𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ ciun 4455  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957   ∧ w-bnj17 30005   predc-bnj14 30007   FrSe w-bnj15 30011   trClc-bnj18 30013 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012 This theorem is referenced by:  bnj1020  30287
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