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Theorem bnj1001 31408
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
bnj1001.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1001.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj1001.6 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
bnj1001.13 𝐷 = (ω ∖ {∅})
bnj1001.27 ((𝜃𝜒𝜏𝜂) → 𝜒″)
Assertion
Ref Expression
bnj1001 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))

Proof of Theorem bnj1001
StepHypRef Expression
1 bnj1001.27 . 2 ((𝜃𝜒𝜏𝜂) → 𝜒″)
2 bnj1001.6 . . . . 5 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
32simplbi 491 . . . 4 (𝜂𝑖𝑛)
43bnj708 31206 . . 3 ((𝜃𝜒𝜏𝜂) → 𝑖𝑛)
5 bnj1001.3 . . . . . 6 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
65bnj1232 31254 . . . . 5 (𝜒𝑛𝐷)
76bnj706 31204 . . . 4 ((𝜃𝜒𝜏𝜂) → 𝑛𝐷)
8 bnj1001.13 . . . . 5 𝐷 = (ω ∖ {∅})
98bnj923 31218 . . . 4 (𝑛𝐷𝑛 ∈ ω)
107, 9syl 17 . . 3 ((𝜃𝜒𝜏𝜂) → 𝑛 ∈ ω)
11 elnn 7273 . . 3 ((𝑖𝑛𝑛 ∈ ω) → 𝑖 ∈ ω)
124, 10, 11syl2anc 579 . 2 ((𝜃𝜒𝜏𝜂) → 𝑖 ∈ ω)
13 bnj1001.5 . . . . . 6 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
1413simp3bi 1177 . . . . 5 (𝜏𝑝 = suc 𝑛)
1514bnj707 31205 . . . 4 ((𝜃𝜒𝜏𝜂) → 𝑝 = suc 𝑛)
16 nnord 7271 . . . . . . 7 (𝑛 ∈ ω → Ord 𝑛)
17 ordsucelsuc 7220 . . . . . . 7 (Ord 𝑛 → (𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
189, 16, 173syl 18 . . . . . 6 (𝑛𝐷 → (𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
1918biimpa 468 . . . . 5 ((𝑛𝐷𝑖𝑛) → suc 𝑖 ∈ suc 𝑛)
20 eleq2 2833 . . . . 5 (𝑝 = suc 𝑛 → (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
2119, 20anim12i 606 . . . 4 (((𝑛𝐷𝑖𝑛) ∧ 𝑝 = suc 𝑛) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
227, 4, 15, 21syl21anc 866 . . 3 ((𝜃𝜒𝜏𝜂) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
23 bianir 1081 . . 3 ((suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)) → suc 𝑖𝑝)
2422, 23syl 17 . 2 ((𝜃𝜒𝜏𝜂) → suc 𝑖𝑝)
251, 12, 243jca 1158 1 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  cdif 3729  c0 4079  {csn 4334  Ord word 5907  suc csuc 5910   Fn wfn 6063  cfv 6068  ωcom 7263  w-bnj17 31135
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 2070  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
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 2063  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-br 4810  df-opab 4872  df-tr 4912  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-om 7264  df-bnj17 31136
This theorem is referenced by:  bnj1020  31413
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