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Theorem bnj557 31309
Description: Technical lemma for bnj852 31329. 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 1093 . . . . 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 31308 . . . . . . 7 (𝜂𝜎)
543anim3i 1157 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → (𝑅 FrSe 𝐴𝜏𝜎))
6 bnj557.20 . . . . . . 7 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
7 vex 3354 . . . . . . . 8 𝑖 ∈ V
87bnj216 31138 . . . . . . 7 (𝑚 = suc 𝑖𝑖𝑚)
96, 8bnj837 31169 . . . . . 6 (𝜁𝑖𝑚)
105, 9anim12i 600 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜁) → ((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚))
111, 10sylbir 225 . . . 4 (((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)) → ((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚))
123bnj1254 31218 . . . . . 6 (𝜂𝑚 = suc 𝑝)
136simp3bi 1141 . . . . . 6 (𝜁𝑚 = suc 𝑖)
14 bnj551 31150 . . . . . 6 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)
1512, 13, 14syl2an 583 . . . . 5 ((𝜂𝜁) → 𝑝 = 𝑖)
1615adantl 467 . . . 4 (((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)) → 𝑝 = 𝑖)
1711, 16jca 501 . . 3 (((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)) → (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) ∧ 𝑝 = 𝑖))
18 bnj256 31112 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) ↔ ((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)))
19 df-3an 1073 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) ↔ (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) ∧ 𝑝 = 𝑖))
2017, 18, 193imtr4i 281 . 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 31306 . 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 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  cdif 3720  cun 3721  c0 4063  {csn 4316  cop 4322   ciun 4654  suc csuc 5868   Fn wfn 6026  cfv 6031  ωcom 7212  w-bnj17 31092   predc-bnj14 31094   FrSe w-bnj15 31098
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 4915  ax-nul 4923  ax-pr 5034  ax-un 7096  ax-reg 8653
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  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 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-id 5157  df-eprel 5162  df-fr 5208  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-bnj17 31093
This theorem is referenced by:  bnj558  31310
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