Theorem bnj571 35519

Description: Technical lemma for bnj852 35534. 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
bnj571.3	⊢ 𝐷 = (ω ∖ {∅})
bnj571.16	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj571.17	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj571.18	⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
bnj571.19	⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj571.20	⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
bnj571.22	⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj571.23	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
bnj571.24	⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj571.25	⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)
bnj571.26	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
bnj571.29	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj571.30	⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj571.38	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)
bnj571.21	⊢ (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
bnj571.40	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝐺 Fn 𝑛)
bnj571.33	⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj571	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜓″)

Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑦,𝐺   𝑅,𝑖,𝑝,𝑦   𝜂,𝑖   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝   𝑖,𝜑′,𝑝

Allowed substitution hints:   𝜏(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜂(𝑥, 𝑦, 𝑓, 𝑚, 𝑛, 𝑝)   𝜁(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜎(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜌(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, 𝑚, 𝑛)   𝜓′(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜓″(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)

Proof of Theorem bnj571

Step	Hyp	Ref	Expression
1		nfv 1947	. . . 4 ⊢ Ⅎ𝑖 𝑅 FrSe 𝐴
2		bnj571.17	. . . . 5 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
3		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑖 𝑓 Fn 𝑚
4		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑖𝜑′
5		bnj571.30	. . . . . . 7 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
6		nfra1 3287	. . . . . . 7 ⊢ Ⅎ𝑖∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
7	5, 6	nfxfr 1886	. . . . . 6 ⊢ Ⅎ𝑖𝜓′
8	3, 4, 7	nf3an 1934	. . . . 5 ⊢ Ⅎ𝑖(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)
9	2, 8	nfxfr 1886	. . . 4 ⊢ Ⅎ𝑖𝜏
10		nfv 1947	. . . 4 ⊢ Ⅎ𝑖𝜂
11	1, 9, 10	nf3an 1934	. . 3 ⊢ Ⅎ𝑖(𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂)
12		df-bnj17 35301	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜁))
13		3anass 1111	. . . . . . . . . 10 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 = suc 𝑖) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 = suc 𝑖)))
14		3anrot 1117	. . . . . . . . . 10 ⊢ ((𝑚 = suc 𝑖 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 = suc 𝑖))
15		bnj571.20	. . . . . . . . . . . 12 ⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
16		df-3an 1105	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖) ↔ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 = suc 𝑖))
17	15, 16	bitri 278	. . . . . . . . . . 11 ⊢ (𝜁 ↔ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 = suc 𝑖))
18	17	anbi2i 635	. . . . . . . . . 10 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜁) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 = suc 𝑖)))
19	13, 14, 18	3bitr4ri 307	. . . . . . . . 9 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜁) ↔ (𝑚 = suc 𝑖 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)))
20	12, 19	bitri 278	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) ↔ (𝑚 = suc 𝑖 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)))
21		bnj571.3	. . . . . . . . 9 ⊢ 𝐷 = (ω ∖ {∅})
22		bnj571.16	. . . . . . . . 9 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
23		bnj571.18	. . . . . . . . 9 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
24		bnj571.19	. . . . . . . . 9 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
25		bnj571.22	. . . . . . . . 9 ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
26		bnj571.23	. . . . . . . . 9 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
27		bnj571.24	. . . . . . . . 9 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
28		bnj571.25	. . . . . . . . 9 ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)
29		bnj571.26	. . . . . . . . 9 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
30		bnj571.29	. . . . . . . . 9 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
31		bnj571.38	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)
32	21, 22, 2, 23, 24, 15, 25, 26, 27, 28, 29, 30, 5, 31	bnj558 35515	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → (𝐺‘suc 𝑖) = 𝐾)
33	20, 32	sylbir 238	. . . . . . 7 ⊢ ((𝑚 = suc 𝑖 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = 𝐾)
34	33	3expib 1140	. . . . . 6 ⊢ (𝑚 = suc 𝑖 → (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = 𝐾))
35		df-bnj17 35301	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜌))
36		3anass 1111	. . . . . . . . . 10 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 ≠ suc 𝑖) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 ≠ suc 𝑖)))
37		3anrot 1117	. . . . . . . . . 10 ⊢ ((𝑚 ≠ suc 𝑖 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 ≠ suc 𝑖))
38		bnj571.21	. . . . . . . . . . . 12 ⊢ (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
39		df-3an 1105	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖) ↔ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 ≠ suc 𝑖))
40	38, 39	bitri 278	. . . . . . . . . . 11 ⊢ (𝜌 ↔ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 ≠ suc 𝑖))
41	40	anbi2i 635	. . . . . . . . . 10 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜌) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 ≠ suc 𝑖)))
42	36, 37, 41	3bitr4ri 307	. . . . . . . . 9 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜌) ↔ (𝑚 ≠ suc 𝑖 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)))
43	35, 42	bitri 278	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) ↔ (𝑚 ≠ suc 𝑖 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)))
44		bnj571.40	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝐺 Fn 𝑛)
45	21, 2, 24, 38, 27, 22, 44, 5	bnj570 35518	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘suc 𝑖) = 𝐾)
46	43, 45	sylbir 238	. . . . . . 7 ⊢ ((𝑚 ≠ suc 𝑖 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = 𝐾)
47	46	3expib 1140	. . . . . 6 ⊢ (𝑚 ≠ suc 𝑖 → (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = 𝐾))
48	34, 47	pm2.61ine 3039	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = 𝐾)
49	48, 27	eqtrdi 2812	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
50	49	exp32 426	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → (𝑖 ∈ ω → (suc 𝑖 ∈ 𝑛 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))))
51	11, 50	alrimi 2250	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → ∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑛 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))))
52		bnj571.33	. . 3 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
53	52	bnj946 35388	. 2 ⊢ (𝜓″ ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑛 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))))
54	51, 53	sylibr 237	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜓″)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077   ∖ cdif 3896   ∪ cun 3897  ∅c0 4279  {csn 4584  ⟨cop 4590  ∪ ciun 4951  suc csuc 6357   Fn wfn 6526  ‘cfv 6531  ωcom 7866   ∧ w-bnj17 35300   predc-bnj14 35302   FrSe w-bnj15 35306

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7740  ax-reg 9570

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-fv 6539  df-om 7867  df-bnj17 35301

This theorem is used by:  bnj600  35532  bnj908  35544