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Theorem bnj557 34915
Description: Technical lemma for bnj852 34935. 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
bnj557.3 𝐷 = (ω ∖ {∅})
bnj557.16 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj557.17 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj557.18 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj557.19 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj557.20 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
bnj557.21 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
bnj557.22 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
bnj557.23 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
bnj557.24 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
bnj557.25 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
bnj557.28 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj557.29 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj557.36 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Assertion
Ref Expression
bnj557 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → (𝐺𝑚) = 𝐿)
Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑦,𝐺   𝑅,𝑖,𝑝,𝑦   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝   𝑝,𝜑′
Allowed substitution hints:   𝜏(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜁(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(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,𝑖,𝑚,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj557
StepHypRef Expression
1 3an4anass 1105 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜁) ↔ ((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)))
2 bnj557.18 . . . . . . . 8 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
3 bnj557.19 . . . . . . . 8 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
42, 3bnj556 34914 . . . . . . 7 (𝜂𝜎)
543anim3i 1155 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → (𝑅 FrSe 𝐴𝜏𝜎))
6 bnj557.20 . . . . . . 7 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
7 vex 3484 . . . . . . . 8 𝑖 ∈ V
87bnj216 34746 . . . . . . 7 (𝑚 = suc 𝑖𝑖𝑚)
96, 8bnj837 34775 . . . . . 6 (𝜁𝑖𝑚)
105, 9anim12i 613 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜁) → ((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚))
111, 10sylbir 235 . . . 4 (((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)) → ((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚))
123bnj1254 34823 . . . . . 6 (𝜂𝑚 = suc 𝑝)
136simp3bi 1148 . . . . . 6 (𝜁𝑚 = suc 𝑖)
14 bnj551 34756 . . . . . 6 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)
1512, 13, 14syl2an 596 . . . . 5 ((𝜂𝜁) → 𝑝 = 𝑖)
1615adantl 481 . . . 4 (((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)) → 𝑝 = 𝑖)
1711, 16jca 511 . . 3 (((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)) → (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) ∧ 𝑝 = 𝑖))
18 bnj256 34720 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) ↔ ((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)))
19 df-3an 1089 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) ↔ (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) ∧ 𝑝 = 𝑖))
2017, 18, 193imtr4i 292 . 2 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → ((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖))
21 bnj557.28 . . 3 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
22 bnj557.29 . . 3 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
23 bnj557.3 . . 3 𝐷 = (ω ∖ {∅})
24 bnj557.16 . . 3 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
25 bnj557.17 . . 3 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
26 bnj557.22 . . 3 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
27 bnj557.25 . . 3 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
28 bnj557.21 . . 3 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
29 bnj557.23 . . 3 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
30 bnj557.24 . . 3 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
31 bnj557.36 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
3221, 22, 23, 24, 25, 2, 26, 27, 28, 29, 30, 31bnj553 34912 . 2 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → (𝐺𝑚) = 𝐿)
3320, 32syl 17 1 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → (𝐺𝑚) = 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cdif 3948  cun 3949  c0 4333  {csn 4626  cop 4632   ciun 4991  suc csuc 6386   Fn wfn 6556  cfv 6561  ωcom 7887  w-bnj17 34700   predc-bnj14 34702   FrSe w-bnj15 34706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-reg 9632
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-id 5578  df-eprel 5584  df-fr 5637  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-bnj17 34701
This theorem is referenced by:  bnj558  34916
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