Theorem bnj558 32167

Description: Technical lemma for bnj852 32186. 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 32166	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → (𝐺‘𝑚) = 𝐿)
16		bnj422 31978	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) ↔ (𝜂 ∧ 𝜁 ∧ 𝑅 FrSe 𝐴 ∧ 𝜏))
17		bnj253 31967	. . . . 5 ⊢ ((𝜂 ∧ 𝜁 ∧ 𝑅 FrSe 𝐴 ∧ 𝜏) ↔ ((𝜂 ∧ 𝜁) ∧ 𝑅 FrSe 𝐴 ∧ 𝜏))
18	16, 17	bitri 277	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) ↔ ((𝜂 ∧ 𝜁) ∧ 𝑅 FrSe 𝐴 ∧ 𝜏))
19	18	simp1bi 1139	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → (𝜂 ∧ 𝜁))
20	5, 6, 9, 10, 9, 10	bnj554 32164	. . 3 ⊢ ((𝜂 ∧ 𝜁) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
21	19, 20	syl 17	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
22	15, 21	mpbid 234	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → (𝐺‘suc 𝑖) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107  ∀wral 3136   ∖ cdif 3931   ∪ cun 3932  ∅c0 4289  {csn 4559  ⟨cop 4565  ∪ ciun 4910  suc csuc 6186   Fn wfn 6343  ‘cfv 6348  ωcom 7572   ∧ w-bnj17 31949   predc-bnj14 31951   FrSe w-bnj15 31955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453  ax-reg 9048

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-id 5453  df-eprel 5458  df-fr 5507  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-bnj17 31950

This theorem is referenced by:  bnj571  32171