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Theorem bnj571 30705
Description: Technical lemma for bnj852 30720. 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 1840 . . . 4 𝑖 𝑅 FrSe 𝐴
2 bnj571.17 . . . . 5 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
3 nfv 1840 . . . . . 6 𝑖 𝑓 Fn 𝑚
4 nfv 1840 . . . . . 6 𝑖𝜑′
5 bnj571.30 . . . . . . 7 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
6 nfra1 2936 . . . . . . 7 𝑖𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
75, 6nfxfr 1776 . . . . . 6 𝑖𝜓′
83, 4, 7nf3an 1828 . . . . 5 𝑖(𝑓 Fn 𝑚𝜑′𝜓′)
92, 8nfxfr 1776 . . . 4 𝑖𝜏
10 nfv 1840 . . . 4 𝑖𝜂
111, 9, 10nf3an 1828 . . 3 𝑖(𝑅 FrSe 𝐴𝜏𝜂)
12 df-bnj17 30481 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) ↔ ((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜁))
13 3anass 1040 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 = suc 𝑖) ↔ ((𝑅 FrSe 𝐴𝜏𝜂) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 = suc 𝑖)))
14 3anrot 1041 . . . . . . . . . 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 1038 . . . . . . . . . . . 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 729 . . . . . . . . . 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 30701 . . . . . . . 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 1265 . . . . . 6 (𝑚 = suc 𝑖 → (((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝐾))
35 df-bnj17 30481 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) ↔ ((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜌))
36 3anass 1040 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 ≠ suc 𝑖) ↔ ((𝑅 FrSe 𝐴𝜏𝜂) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝑚 ≠ suc 𝑖)))
37 3anrot 1041 . . . . . . . . . 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 1038 . . . . . . . . . . . 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 729 . . . . . . . . . 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 30704 . . . . . . . 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 1265 . . . . . 6 (𝑚 ≠ suc 𝑖 → (((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝐾))
4834, 47pm2.61ine 2873 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝐾)
4948, 27syl6eq 2671 . . . 4 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
5049exp32 630 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂) → (𝑖 ∈ ω → (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))))
5111, 50alrimi 2080 . 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 30574 . 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 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wne 2790  wral 2907  cdif 3553  cun 3554  c0 3893  {csn 4150  cop 4156   ciun 4487  suc csuc 5686   Fn wfn 5844  cfv 5849  ωcom 7015  w-bnj17 30480   predc-bnj14 30482   FrSe w-bnj15 30486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869  ax-un 6905  ax-reg 8444
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-res 5088  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-fv 5857  df-om 7016  df-bnj17 30481
This theorem is referenced by:  bnj600  30718  bnj908  30730
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