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Theorem bnj571 31315
Description: Technical lemma for bnj852 31330. 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
bnj571.3 𝐷 = (ω ∖ {∅})
bnj571.16 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj571.17 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj571.18 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj571.19 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj571.20 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
bnj571.22 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
bnj571.23 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
bnj571.24 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
bnj571.25 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
bnj571.26 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
bnj571.29 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj571.30 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj571.38 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
bnj571.21 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
bnj571.40 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
bnj571.33 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj571 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜓″)
Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑦,𝐺   𝑅,𝑖,𝑝,𝑦   𝜂,𝑖   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝   𝑖,𝜑′,𝑝
Allowed substitution hints:   𝜏(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑥,𝑦,𝑓,𝑚,𝑛,𝑝)   𝜁(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜌(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑥,𝑓,𝑚,𝑛)   𝐵(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑥,𝑓,𝑚,𝑛)   𝐺(𝑥,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐾(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐿(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑′(𝑥,𝑦,𝑓,𝑚,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj571
StepHypRef Expression
1 nfv 1995 . . . 4 𝑖 𝑅 FrSe 𝐴
2 bnj571.17 . . . . 5 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
3 nfv 1995 . . . . . 6 𝑖 𝑓 Fn 𝑚
4 nfv 1995 . . . . . 6 𝑖𝜑′
5 bnj571.30 . . . . . . 7 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
6 nfra1 3090 . . . . . . 7 𝑖𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
75, 6nfxfr 1929 . . . . . 6 𝑖𝜓′
83, 4, 7nf3an 1983 . . . . 5 𝑖(𝑓 Fn 𝑚𝜑′𝜓′)
92, 8nfxfr 1929 . . . 4 𝑖𝜏
10 nfv 1995 . . . 4 𝑖𝜂
111, 9, 10nf3an 1983 . . 3 𝑖(𝑅 FrSe 𝐴𝜏𝜂)
12 df-bnj17 31094 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) ↔ ((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜁))
13 3anass 1080 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 = suc 𝑖) ↔ ((𝑅 FrSe 𝐴𝜏𝜂) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 = suc 𝑖)))
14 3anrot 1086 . . . . . . . . . 10 ((𝑚 = suc 𝑖 ∧ (𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)) ↔ ((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 = suc 𝑖))
15 bnj571.20 . . . . . . . . . . . 12 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
16 df-3an 1073 . . . . . . . . . . . 12 ((𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖) ↔ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 = suc 𝑖))
1715, 16bitri 264 . . . . . . . . . . 11 (𝜁 ↔ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 = suc 𝑖))
1817anbi2i 603 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜁) ↔ ((𝑅 FrSe 𝐴𝜏𝜂) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 = suc 𝑖)))
1913, 14, 183bitr4ri 293 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜁) ↔ (𝑚 = suc 𝑖 ∧ (𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)))
2012, 19bitri 264 . . . . . . . 8 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) ↔ (𝑚 = suc 𝑖 ∧ (𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)))
21 bnj571.3 . . . . . . . . 9 𝐷 = (ω ∖ {∅})
22 bnj571.16 . . . . . . . . 9 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
23 bnj571.18 . . . . . . . . 9 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
24 bnj571.19 . . . . . . . . 9 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
25 bnj571.22 . . . . . . . . 9 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
26 bnj571.23 . . . . . . . . 9 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
27 bnj571.24 . . . . . . . . 9 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
28 bnj571.25 . . . . . . . . 9 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
29 bnj571.26 . . . . . . . . 9 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
30 bnj571.29 . . . . . . . . 9 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
31 bnj571.38 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
3221, 22, 2, 23, 24, 15, 25, 26, 27, 28, 29, 30, 5, 31bnj558 31311 . . . . . . . 8 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → (𝐺‘suc 𝑖) = 𝐾)
3320, 32sylbir 225 . . . . . . 7 ((𝑚 = suc 𝑖 ∧ (𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝐾)
34333expib 1116 . . . . . 6 (𝑚 = suc 𝑖 → (((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝐾))
35 df-bnj17 31094 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) ↔ ((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜌))
36 3anass 1080 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 ≠ suc 𝑖) ↔ ((𝑅 FrSe 𝐴𝜏𝜂) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 ≠ suc 𝑖)))
37 3anrot 1086 . . . . . . . . . 10 ((𝑚 ≠ suc 𝑖 ∧ (𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)) ↔ ((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 ≠ suc 𝑖))
38 bnj571.21 . . . . . . . . . . . 12 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
39 df-3an 1073 . . . . . . . . . . . 12 ((𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖) ↔ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 ≠ suc 𝑖))
4038, 39bitri 264 . . . . . . . . . . 11 (𝜌 ↔ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 ≠ suc 𝑖))
4140anbi2i 603 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜌) ↔ ((𝑅 FrSe 𝐴𝜏𝜂) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 ≠ suc 𝑖)))
4236, 37, 413bitr4ri 293 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜌) ↔ (𝑚 ≠ suc 𝑖 ∧ (𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)))
4335, 42bitri 264 . . . . . . . 8 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) ↔ (𝑚 ≠ suc 𝑖 ∧ (𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)))
44 bnj571.40 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
4521, 2, 24, 38, 27, 22, 44, 5bnj570 31314 . . . . . . . 8 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = 𝐾)
4643, 45sylbir 225 . . . . . . 7 ((𝑚 ≠ suc 𝑖 ∧ (𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝐾)
47463expib 1116 . . . . . 6 (𝑚 ≠ suc 𝑖 → (((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝐾))
4834, 47pm2.61ine 3026 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝐾)
4948, 27syl6eq 2821 . . . 4 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
5049exp32 407 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂) → (𝑖 ∈ ω → (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))))
5111, 50alrimi 2238 . 2 ((𝑅 FrSe 𝐴𝜏𝜂) → ∀𝑖(𝑖 ∈ ω → (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))))
52 bnj571.33 . . 3 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
5352bnj946 31184 . 2 (𝜓″ ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))))
5451, 53sylibr 224 1 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜓″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071  wal 1629   = wceq 1631  wcel 2145  wne 2943  wral 3061  cdif 3721  cun 3722  c0 4064  {csn 4317  cop 4323   ciun 4655  suc csuc 5869   Fn wfn 6027  cfv 6032  ωcom 7213  w-bnj17 31093   predc-bnj14 31095   FrSe w-bnj15 31099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7097  ax-reg 8654
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-res 5262  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-fv 6040  df-om 7214  df-bnj17 31094
This theorem is referenced by:  bnj600  31328  bnj908  31340
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