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Theorem bnj967 31141
Description: Technical lemma for bnj69 31204. 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 1101 . . . . . 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 31001 . . . . . . . 8 (𝜒𝑓 Fn 𝑛)
543ad2ant1 1102 . . . . . . 7 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑓 Fn 𝑛)
653ad2ant2 1103 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → 𝑓 Fn 𝑛)
7 simp23 1116 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → 𝑝 = suc 𝑛)
8 simp3 1083 . . . . . . 7 ((𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛) → suc 𝑖𝑛)
983ad2ant3 1104 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → suc 𝑖𝑛)
102, 6, 7, 9bnj951 30972 . . . . 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 30964 . . . . . . . . 9 (𝑛𝐷𝑛 ∈ ω)
133, 12bnj769 30958 . . . . . . . 8 (𝜒𝑛 ∈ ω)
14133ad2ant1 1102 . . . . . . 7 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑛 ∈ ω)
1514, 8bnj240 30893 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝑛 ∈ ω ∧ suc 𝑖𝑛))
16 nnord 7115 . . . . . . . 8 (𝑛 ∈ ω → Ord 𝑛)
17 ordtr 5775 . . . . . . . 8 (Ord 𝑛 → Tr 𝑛)
1816, 17syl 17 . . . . . . 7 (𝑛 ∈ ω → Tr 𝑛)
19 trsuc 5848 . . . . . . 7 ((Tr 𝑛 ∧ suc 𝑖𝑛) → 𝑖𝑛)
2018, 19sylan 487 . . . . . 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 30947 . . . . . . 7 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛 ∧ suc 𝑖𝑛) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛))
2322anim1i 591 . . . . . 6 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛 ∧ suc 𝑖𝑛) ∧ 𝑖𝑛) → ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝑖𝑛))
24 df-bnj17 30881 . . . . . 6 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝑖𝑛))
2523, 24sylibr 224 . . . . 5 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛 ∧ suc 𝑖𝑛) ∧ 𝑖𝑛) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛))
2610, 21, 25syl2anc 694 . . . 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 30970 . . . 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 30970 . . . 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 1079 . . . 4 ((𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛) → (𝑖 ∈ ω ∧ suc 𝑖𝑛))
33323ad2ant3 1104 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝑖 ∈ ω ∧ suc 𝑖𝑛))
343bnj1254 31006 . . . . 5 (𝜒𝜓)
35343ad2ant1 1102 . . . 4 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝜓)
36353ad2ant2 1103 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → 𝜓)
3729, 31, 33, 36bnj951 30972 . 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 31136 . . 3 ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))
4138, 40bnj953 31135 . 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cdif 3604  cun 3605  c0 3948  {csn 4210  cop 4216   ciun 4552  Tr wtr 4785  Ord word 5760  suc csuc 5763   Fn wfn 5921  cfv 5926  ωcom 7107  w-bnj17 30880   predc-bnj14 30882   FrSe w-bnj15 30886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-om 7108  df-bnj17 30881
This theorem is referenced by:  bnj910  31144
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