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Theorem bnj967 31395
Description: Technical lemma for bnj69 31458. 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
bnj967.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj967.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj967.10 𝐷 = (ω ∖ {∅})
bnj967.12 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj967.13 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj967.44 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
Assertion
Ref Expression
bnj967 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
Distinct variable groups:   𝑦,𝑓   𝑦,𝑖   𝑦,𝑛
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐺(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj967
StepHypRef Expression
1 bnj967.44 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
213adant3 1162 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → 𝐶 ∈ V)
3 bnj967.3 . . . . . . . . 9 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
43bnj1235 31255 . . . . . . . 8 (𝜒𝑓 Fn 𝑛)
543ad2ant1 1163 . . . . . . 7 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑓 Fn 𝑛)
653ad2ant2 1164 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → 𝑓 Fn 𝑛)
7 simp23 1265 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → 𝑝 = suc 𝑛)
8 simp3 1168 . . . . . . 7 ((𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛) → suc 𝑖𝑛)
983ad2ant3 1165 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → suc 𝑖𝑛)
102, 6, 7, 9bnj951 31226 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛 ∧ suc 𝑖𝑛))
11 bnj967.10 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
1211bnj923 31218 . . . . . . . . 9 (𝑛𝐷𝑛 ∈ ω)
133, 12bnj769 31212 . . . . . . . 8 (𝜒𝑛 ∈ ω)
14133ad2ant1 1163 . . . . . . 7 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑛 ∈ ω)
1514, 8bnj240 31148 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝑛 ∈ ω ∧ suc 𝑖𝑛))
16 nnord 7271 . . . . . . . 8 (𝑛 ∈ ω → Ord 𝑛)
17 ordtr 5922 . . . . . . . 8 (Ord 𝑛 → Tr 𝑛)
1816, 17syl 17 . . . . . . 7 (𝑛 ∈ ω → Tr 𝑛)
19 trsuc 5992 . . . . . . 7 ((Tr 𝑛 ∧ suc 𝑖𝑛) → 𝑖𝑛)
2018, 19sylan 575 . . . . . 6 ((𝑛 ∈ ω ∧ suc 𝑖𝑛) → 𝑖𝑛)
2115, 20syl 17 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → 𝑖𝑛)
22 bnj658 31201 . . . . . . 7 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛 ∧ suc 𝑖𝑛) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛))
2322anim1i 608 . . . . . 6 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛 ∧ suc 𝑖𝑛) ∧ 𝑖𝑛) → ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝑖𝑛))
24 df-bnj17 31136 . . . . . 6 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝑖𝑛))
2523, 24sylibr 225 . . . . 5 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛 ∧ suc 𝑖𝑛) ∧ 𝑖𝑛) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛))
2610, 21, 25syl2anc 579 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛))
27 bnj967.13 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
2827bnj945 31224 . . . 4 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛) → (𝐺𝑖) = (𝑓𝑖))
2926, 28syl 17 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐺𝑖) = (𝑓𝑖))
3027bnj945 31224 . . . 4 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛 ∧ suc 𝑖𝑛) → (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖))
3110, 30syl 17 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖))
32 3simpb 1180 . . . 4 ((𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛) → (𝑖 ∈ ω ∧ suc 𝑖𝑛))
33323ad2ant3 1165 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝑖 ∈ ω ∧ suc 𝑖𝑛))
343bnj1254 31260 . . . . 5 (𝜒𝜓)
35343ad2ant1 1163 . . . 4 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝜓)
36353ad2ant2 1164 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → 𝜓)
3729, 31, 33, 36bnj951 31226 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → ((𝐺𝑖) = (𝑓𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓))
38 bnj967.2 . . 3 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
39 bnj967.12 . . . 4 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
4039, 27bnj958 31390 . . 3 ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))
4138, 40bnj953 31389 . 2 (((𝐺𝑖) = (𝑓𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
4237, 41syl 17 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cdif 3729  cun 3730  c0 4079  {csn 4334  cop 4340   ciun 4676  Tr wtr 4911  Ord word 5907  suc csuc 5910   Fn wfn 6063  cfv 6068  ωcom 7263  w-bnj17 31135   predc-bnj14 31137   FrSe w-bnj15 31141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147  ax-reg 8704
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-res 5289  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-fv 6076  df-om 7264  df-bnj17 31136
This theorem is referenced by:  bnj910  31398
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