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Theorem bnj998 32114
 Description: Technical lemma for bnj69 32166. 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 31860 . . . . . . 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 218 . . . . 5 (𝜃 → (𝑅 FrSe 𝐴𝑋𝐴))
54bnj705 31910 . . . 4 ((𝜃𝜒𝜏𝜂) → (𝑅 FrSe 𝐴𝑋𝐴))
6 bnj643 31906 . . . 4 ((𝜃𝜒𝜏𝜂) → 𝜒)
7 bnj998.5 . . . . . 6 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
8 3simpc 1144 . . . . . 6 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
97, 8sylbi 218 . . . . 5 (𝜏 → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
109bnj707 31912 . . . 4 ((𝜃𝜒𝜏𝜂) → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
11 bnj255 31861 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜒 ∧ (𝑛 = suc 𝑚𝑝 = suc 𝑛)))
125, 6, 10, 11syl3anbrc 1337 . . 3 ((𝜃𝜒𝜏𝜂) → ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛))
13 bnj252 31859 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)))
1412, 13sylib 219 . 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 262 . . 3 ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓))
29 biid 262 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑚𝑛) ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
3015, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29bnj910 32106 . 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 207   ∧ wa 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107  {cab 2804  ∀wral 3143  ∃wrex 3144  [wsbc 3776   ∖ cdif 3937   ∪ cun 3938  ∅c0 4295  {csn 4564  ⟨cop 4570  ∪ ciun 4917  suc csuc 6191   Fn wfn 6347  ‘cfv 6352  ωcom 7568   ∧ w-bnj17 31842   predc-bnj14 31844   FrSe w-bnj15 31848   trClc-bnj18 31850 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7451  ax-reg 9045 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-om 7569  df-bnj17 31843  df-bnj14 31845  df-bnj13 31847  df-bnj15 31849 This theorem is referenced by:  bnj1020  32121
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