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Theorem bnj558 31310
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
bnj558.3 𝐷 = (ω ∖ {∅})
bnj558.16 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj558.17 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj558.18 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj558.19 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj558.20 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
bnj558.21 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
bnj558.22 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
bnj558.23 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
bnj558.24 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
bnj558.25 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
bnj558.28 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj558.29 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj558.36 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Assertion
Ref Expression
bnj558 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → (𝐺‘suc 𝑖) = 𝐾)
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 bnj558
StepHypRef Expression
1 bnj558.3 . . 3 𝐷 = (ω ∖ {∅})
2 bnj558.16 . . 3 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
3 bnj558.17 . . 3 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
4 bnj558.18 . . 3 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
5 bnj558.19 . . 3 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
6 bnj558.20 . . 3 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
7 bnj558.21 . . 3 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
8 bnj558.22 . . 3 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
9 bnj558.23 . . 3 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
10 bnj558.24 . . 3 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
11 bnj558.25 . . 3 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
12 bnj558.28 . . 3 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
13 bnj558.29 . . 3 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
14 bnj558.36 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14bnj557 31309 . 2 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → (𝐺𝑚) = 𝐿)
16 bnj422 31121 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) ↔ (𝜂𝜁𝑅 FrSe 𝐴𝜏))
17 bnj253 31110 . . . . 5 ((𝜂𝜁𝑅 FrSe 𝐴𝜏) ↔ ((𝜂𝜁) ∧ 𝑅 FrSe 𝐴𝜏))
1816, 17bitri 264 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) ↔ ((𝜂𝜁) ∧ 𝑅 FrSe 𝐴𝜏))
1918simp1bi 1139 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → (𝜂𝜁))
205, 6, 9, 10, 9, 10bnj554 31307 . . 3 ((𝜂𝜁) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
2119, 20syl 17 . 2 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
2215, 21mpbid 222 1 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → (𝐺‘suc 𝑖) = 𝐾)
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 4317  cop 4323   ciun 4655  suc csuc 5867   Fn wfn 6025  cfv 6030  ωcom 7216  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 4916  ax-nul 4924  ax-pr 5035  ax-un 7100  ax-reg 8657
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-id 5158  df-eprel 5163  df-fr 5209  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-res 5262  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-fv 6038  df-bnj17 31093
This theorem is referenced by:  bnj571  31314
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