Theorem bnj558 35599

Description: Technical lemma for bnj852 35618. 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 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj558.17	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj558.18	⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
bnj558.19	⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj558.20	⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
bnj558.21	⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj558.22	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
bnj558.23	⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj558.24	⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)
bnj558.25	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
bnj558.28	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj558.29	⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj558.36	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)

Assertion

Ref	Expression
bnj558	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → (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

Step	Hyp	Ref	Expression
1		bnj558.3	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
2		bnj558.16	. . 3 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
3		bnj558.17	. . 3 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
4		bnj558.18	. . 3 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
5		bnj558.19	. . 3 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
6		bnj558.20	. . 3 ⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
7		bnj558.21	. . 3 ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
8		bnj558.22	. . 3 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
9		bnj558.23	. . 3 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
10		bnj558.24	. . 3 ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) 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 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
14		bnj558.36	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	bnj557 35598	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → (𝐺‘𝑚) = 𝐿)
16		bnj422 35413	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) ↔ (𝜂 ∧ 𝜁 ∧ 𝑅 FrSe 𝐴 ∧ 𝜏))
17		bnj253 35402	. . . . 5 ⊢ ((𝜂 ∧ 𝜁 ∧ 𝑅 FrSe 𝐴 ∧ 𝜏) ↔ ((𝜂 ∧ 𝜁) ∧ 𝑅 FrSe 𝐴 ∧ 𝜏))
18	16, 17	bitri 278	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) ↔ ((𝜂 ∧ 𝜁) ∧ 𝑅 FrSe 𝐴 ∧ 𝜏))
19	18	simp1bi 1163	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → (𝜂 ∧ 𝜁))
20	5, 6, 9, 10, 9, 10	bnj554 35596	. . 3 ⊢ ((𝜂 ∧ 𝜁) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
21	19, 20	syl 18	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
22	15, 21	mpbid 235	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → (𝐺‘suc 𝑖) = 𝐾)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ∖ cdif 4048   ∪ cun 4152  ∅c0 4281  {csn 4588  ⟨cop 4838  ∪ ciun 4954  suc csuc 6379   Fn wfn 6550  ‘cfv 6555  ωcom 7898   ∧ w-bnj17 35384   predc-bnj14 35386   FrSe w-bnj15 35390

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-un 7772  ax-reg 9610

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-id 5550  df-eprel 5555  df-fr 5608  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-res 5667  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-fv 6563  df-bnj17 35385

This theorem is used by:  bnj571  35603