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Theorem bnj1001 32341
Description: Technical lemma for bnj69 32392. 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 501 . . . 4 (𝜂𝑖𝑛)
43bnj708 32137 . . 3 ((𝜃𝜒𝜏𝜂) → 𝑖𝑛)
5 bnj1001.3 . . . . . 6 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
65bnj1232 32185 . . . . 5 (𝜒𝑛𝐷)
76bnj706 32135 . . . 4 ((𝜃𝜒𝜏𝜂) → 𝑛𝐷)
8 bnj1001.13 . . . . 5 𝐷 = (ω ∖ {∅})
98bnj923 32149 . . . 4 (𝑛𝐷𝑛 ∈ ω)
107, 9syl 17 . . 3 ((𝜃𝜒𝜏𝜂) → 𝑛 ∈ ω)
11 elnn 7570 . . 3 ((𝑖𝑛𝑛 ∈ ω) → 𝑖 ∈ ω)
124, 10, 11syl2anc 587 . 2 ((𝜃𝜒𝜏𝜂) → 𝑖 ∈ ω)
13 bnj1001.5 . . . . . 6 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
1413simp3bi 1144 . . . . 5 (𝜏𝑝 = suc 𝑛)
1514bnj707 32136 . . . 4 ((𝜃𝜒𝜏𝜂) → 𝑝 = suc 𝑛)
16 nnord 7568 . . . . . . 7 (𝑛 ∈ ω → Ord 𝑛)
17 ordsucelsuc 7517 . . . . . . 7 (Ord 𝑛 → (𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
189, 16, 173syl 18 . . . . . 6 (𝑛𝐷 → (𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑛))
1918biimpa 480 . . . . 5 ((𝑛𝐷𝑖𝑛) → suc 𝑖 ∈ suc 𝑛)
20 eleq2 2878 . . . . 5 (𝑝 = suc 𝑛 → (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
2119, 20anim12i 615 . . . 4 (((𝑛𝐷𝑖𝑛) ∧ 𝑝 = suc 𝑛) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
227, 4, 15, 21syl21anc 836 . . 3 ((𝜃𝜒𝜏𝜂) → (suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)))
23 bianir 1054 . . 3 ((suc 𝑖 ∈ suc 𝑛 ∧ (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛)) → suc 𝑖𝑝)
2422, 23syl 17 . 2 ((𝜃𝜒𝜏𝜂) → suc 𝑖𝑝)
251, 12, 243jca 1125 1 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  cdif 3878  c0 4243  {csn 4525  Ord word 6158  suc csuc 6161   Fn wfn 6319  cfv 6324  ωcom 7560  w-bnj17 32066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-om 7561  df-bnj17 32067
This theorem is referenced by:  bnj1020  32347
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