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Theorem bnj557 32247
Description: Technical lemma for bnj852 32267. 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 1102 . . . . 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 32246 . . . . . . 7 (𝜂𝜎)
543anim3i 1151 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → (𝑅 FrSe 𝐴𝜏𝜎))
6 bnj557.20 . . . . . . 7 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
7 vex 3472 . . . . . . . 8 𝑖 ∈ V
87bnj216 32076 . . . . . . 7 (𝑚 = suc 𝑖𝑖𝑚)
96, 8bnj837 32106 . . . . . 6 (𝜁𝑖𝑚)
105, 9anim12i 615 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜁) → ((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚))
111, 10sylbir 238 . . . 4 (((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)) → ((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚))
123bnj1254 32155 . . . . . 6 (𝜂𝑚 = suc 𝑝)
136simp3bi 1144 . . . . . 6 (𝜁𝑚 = suc 𝑖)
14 bnj551 32087 . . . . . 6 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)
1512, 13, 14syl2an 598 . . . . 5 ((𝜂𝜁) → 𝑝 = 𝑖)
1615adantl 485 . . . 4 (((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)) → 𝑝 = 𝑖)
1711, 16jca 515 . . 3 (((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)) → (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) ∧ 𝑝 = 𝑖))
18 bnj256 32050 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) ↔ ((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)))
19 df-3an 1086 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) ↔ (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) ∧ 𝑝 = 𝑖))
2017, 18, 193imtr4i 295 . 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 32244 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wral 3130  cdif 3905  cun 3906  c0 4265  {csn 4539  cop 4545   ciun 4894  suc csuc 6171   Fn wfn 6329  cfv 6334  ωcom 7565  w-bnj17 32030   predc-bnj14 32032   FrSe w-bnj15 32036
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446  ax-reg 9044
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-id 5437  df-eprel 5442  df-fr 5491  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-res 5544  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342  df-bnj17 32031
This theorem is referenced by:  bnj558  32248
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