Theorem bnj571 30976

Description: Technical lemma for bnj852 30991. 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 1843	. . . 4 ⊢ Ⅎ𝑖 𝑅 FrSe 𝐴
2		bnj571.17	. . . . 5 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
3		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑖 𝑓 Fn 𝑚
4		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑖𝜑′
5		bnj571.30	. . . . . . 7 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
6		nfra1 2941	. . . . . . 7 ⊢ Ⅎ𝑖∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
7	5, 6	nfxfr 1779	. . . . . 6 ⊢ Ⅎ𝑖𝜓′
8	3, 4, 7	nf3an 1831	. . . . 5 ⊢ Ⅎ𝑖(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)
9	2, 8	nfxfr 1779	. . . 4 ⊢ Ⅎ𝑖𝜏
10		nfv 1843	. . . 4 ⊢ Ⅎ𝑖𝜂
11	1, 9, 10	nf3an 1831	. . 3 ⊢ Ⅎ𝑖(𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂)
12		df-bnj17 30753	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜁))
13		3anass 1042	. . . . . . . . . 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 1043	. . . . . . . . . 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 1039	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖) ↔ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 = suc 𝑖))
17	15, 16	bitri 264	. . . . . . . . . . 11 ⊢ (𝜁 ↔ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 = suc 𝑖))
18	17	anbi2i 730	. . . . . . . . . 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 293	. . . . . . . . 9 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜁) ↔ (𝑚 = suc 𝑖 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)))
20	12, 19	bitri 264	. . . . . . . 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 30972	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → (𝐺‘suc 𝑖) = 𝐾)
33	20, 32	sylbir 225	. . . . . . 7 ⊢ ((𝑚 = suc 𝑖 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = 𝐾)
34	33	3expib 1268	. . . . . 6 ⊢ (𝑚 = suc 𝑖 → (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = 𝐾))
35		df-bnj17 30753	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜌))
36		3anass 1042	. . . . . . . . . 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 1043	. . . . . . . . . 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 1039	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖) ↔ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 ≠ suc 𝑖))
40	38, 39	bitri 264	. . . . . . . . . . 11 ⊢ (𝜌 ↔ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑚 ≠ suc 𝑖))
41	40	anbi2i 730	. . . . . . . . . 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 293	. . . . . . . . 9 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜌) ↔ (𝑚 ≠ suc 𝑖 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)))
43	35, 42	bitri 264	. . . . . . . 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 30975	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘suc 𝑖) = 𝐾)
46	43, 45	sylbir 225	. . . . . . 7 ⊢ ((𝑚 ≠ suc 𝑖 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = 𝐾)
47	46	3expib 1268	. . . . . 6 ⊢ (𝑚 ≠ suc 𝑖 → (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = 𝐾))
48	34, 47	pm2.61ine 2877	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = 𝐾)
49	48, 27	syl6eq 2672	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
50	49	exp32 631	. . 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 2082	. 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 30845	. 2 ⊢ (𝜓″ ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑛 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))))
54	51, 53	sylibr 224	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜓″)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912   ∖ cdif 3571   ∪ cun 3572  ∅c0 3915  {csn 4177  ⟨cop 4183  ∪ ciun 4520  suc csuc 5725   Fn wfn 5883  ‘cfv 5888  ωcom 7065   ∧ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-om 7066  df-bnj17 30753

This theorem is referenced by:  bnj600  30989  bnj908  31001