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Theorem bnj998 31365
Description: Technical lemma for bnj69 31417. 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
bnj998.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj998.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj998.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj998.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj998.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj998.7 (𝜑′[𝑝 / 𝑛]𝜑)
bnj998.8 (𝜓′[𝑝 / 𝑛]𝜓)
bnj998.9 (𝜒′[𝑝 / 𝑛]𝜒)
bnj998.10 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj998.11 (𝜓″[𝐺 / 𝑓]𝜓′)
bnj998.12 (𝜒″[𝐺 / 𝑓]𝜒′)
bnj998.13 𝐷 = (ω ∖ {∅})
bnj998.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj998.15 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj998.16 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
Assertion
Ref Expression
bnj998 ((𝜃𝜒𝜏𝜂) → 𝜒″)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐷,𝑓,𝑖,𝑛   𝑖,𝐺   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛   𝑓,𝑝,𝑖,𝑛   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,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)   𝐵(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑚,𝑝)   𝑅(𝑧,𝑝)   𝐺(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑧,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 bnj998
StepHypRef Expression
1 bnj998.4 . . . . . 6 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
2 bnj253 31111 . . . . . . 7 ((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)))
32simp1bi 1139 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → (𝑅 FrSe 𝐴𝑋𝐴))
41, 3sylbi 207 . . . . 5 (𝜃 → (𝑅 FrSe 𝐴𝑋𝐴))
54bnj705 31162 . . . 4 ((𝜃𝜒𝜏𝜂) → (𝑅 FrSe 𝐴𝑋𝐴))
6 bnj643 31158 . . . 4 ((𝜃𝜒𝜏𝜂) → 𝜒)
7 bnj998.5 . . . . . 6 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
8 3simpc 1146 . . . . . 6 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
97, 8sylbi 207 . . . . 5 (𝜏 → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
109bnj707 31164 . . . 4 ((𝜃𝜒𝜏𝜂) → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
11 bnj255 31112 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜒 ∧ (𝑛 = suc 𝑚𝑝 = suc 𝑛)))
125, 6, 10, 11syl3anbrc 1428 . . 3 ((𝜃𝜒𝜏𝜂) → ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛))
13 bnj252 31110 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)))
1412, 13sylib 208 . 2 ((𝜃𝜒𝜏𝜂) → ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)))
15 bnj998.1 . . 3 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
16 bnj998.2 . . 3 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
17 bnj998.3 . . 3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
18 bnj998.7 . . 3 (𝜑′[𝑝 / 𝑛]𝜑)
19 bnj998.8 . . 3 (𝜓′[𝑝 / 𝑛]𝜓)
20 bnj998.9 . . 3 (𝜒′[𝑝 / 𝑛]𝜒)
21 bnj998.10 . . 3 (𝜑″[𝐺 / 𝑓]𝜑′)
22 bnj998.11 . . 3 (𝜓″[𝐺 / 𝑓]𝜓′)
23 bnj998.12 . . 3 (𝜒″[𝐺 / 𝑓]𝜒′)
24 bnj998.13 . . 3 𝐷 = (ω ∖ {∅})
25 bnj998.14 . . 3 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
26 bnj998.15 . . 3 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
27 bnj998.16 . . 3 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
28 biid 251 . . 3 ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓))
29 biid 251 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑚𝑛) ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
3015, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29bnj910 31357 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜒″)
3114, 30syl 17 1 ((𝜃𝜒𝜏𝜂) → 𝜒″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  {cab 2757  wral 3061  wrex 3062  [wsbc 3588  cdif 3721  cun 3722  c0 4064  {csn 4317  cop 4323   ciun 4655  suc csuc 5869   Fn wfn 6027  cfv 6032  ωcom 7213  w-bnj17 31093   predc-bnj14 31095   FrSe w-bnj15 31099   trClc-bnj18 31101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7097  ax-reg 8654
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-om 7214  df-bnj17 31094  df-bnj14 31096  df-bnj13 31098  df-bnj15 31100
This theorem is referenced by:  bnj1020  31372
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