Theorem bnj966 34251

Description: Technical lemma for bnj69 34317. 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
bnj966.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj966.10	⊢ 𝐷 = (ω ∖ {∅})
bnj966.12	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj966.13	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj966.44	⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V)
bnj966.53	⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐺 Fn 𝑝)

Assertion

Ref	Expression
bnj966	⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))

Distinct variable groups:   𝑦,𝑓   𝑦,𝑖   𝑦,𝑚   𝑦,𝑛

Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐺(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj966

Step	Hyp	Ref	Expression
1		bnj966.53	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐺 Fn 𝑝)
2	1	fnfund 6651	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → Fun 𝐺)
3	2	3adant3 1130	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → Fun 𝐺)
4		opex 5465	. . . . . . 7 ⊢ ⟨𝑛, 𝐶⟩ ∈ V
5	4	snid 4665	. . . . . 6 ⊢ ⟨𝑛, 𝐶⟩ ∈ {⟨𝑛, 𝐶⟩}
6		elun2 4178	. . . . . 6 ⊢ (⟨𝑛, 𝐶⟩ ∈ {⟨𝑛, 𝐶⟩} → ⟨𝑛, 𝐶⟩ ∈ (𝑓 ∪ {⟨𝑛, 𝐶⟩}))
7	5, 6	ax-mp 5	. . . . 5 ⊢ ⟨𝑛, 𝐶⟩ ∈ (𝑓 ∪ {⟨𝑛, 𝐶⟩})
8		bnj966.13	. . . . 5 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
9	7, 8	eleqtrri 2830	. . . 4 ⊢ ⟨𝑛, 𝐶⟩ ∈ 𝐺
10		funopfv 6944	. . . 4 ⊢ (Fun 𝐺 → (⟨𝑛, 𝐶⟩ ∈ 𝐺 → (𝐺‘𝑛) = 𝐶))
11	3, 9, 10	mpisyl 21	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝐺‘𝑛) = 𝐶)
12		simp22 1205	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝑛 = suc 𝑚)
13		simp33 1209	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝑛 = suc 𝑖)
14		bnj551 34049	. . . . 5 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑛 = suc 𝑖) → 𝑚 = 𝑖)
15	12, 13, 14	syl2anc 582	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝑚 = 𝑖)
16		suceq 6431	. . . . . . . 8 ⊢ (𝑚 = 𝑖 → suc 𝑚 = suc 𝑖)
17	16	eqeq2d 2741	. . . . . . 7 ⊢ (𝑚 = 𝑖 → (𝑛 = suc 𝑚 ↔ 𝑛 = suc 𝑖))
18	17	biimpac 477	. . . . . 6 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑚 = 𝑖) → 𝑛 = suc 𝑖)
19	18	fveq2d 6896	. . . . 5 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑚 = 𝑖) → (𝐺‘𝑛) = (𝐺‘suc 𝑖))
20		bnj966.12	. . . . . . 7 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
21		fveq2 6892	. . . . . . . 8 ⊢ (𝑚 = 𝑖 → (𝑓‘𝑚) = (𝑓‘𝑖))
22	21	bnj1113 34092	. . . . . . 7 ⊢ (𝑚 = 𝑖 → ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
23	20, 22	eqtrid 2782	. . . . . 6 ⊢ (𝑚 = 𝑖 → 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
24	23	adantl 480	. . . . 5 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑚 = 𝑖) → 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
25	19, 24	eqeq12d 2746	. . . 4 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑚 = 𝑖) → ((𝐺‘𝑛) = 𝐶 ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
26	12, 15, 25	syl2anc 582	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → ((𝐺‘𝑛) = 𝐶 ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
27	11, 26	mpbid 231	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
28		bnj966.44	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V)
29	28	3adant3 1130	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝐶 ∈ V)
30		bnj966.3	. . . . . . . 8 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
31	30	bnj1235 34111	. . . . . . 7 ⊢ (𝜒 → 𝑓 Fn 𝑛)
32	31	3ad2ant1 1131	. . . . . 6 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑓 Fn 𝑛)
33	32	3ad2ant2 1132	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝑓 Fn 𝑛)
34		simp23 1206	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝑝 = suc 𝑛)
35	29, 33, 34, 13	bnj951 34082	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑛 = suc 𝑖))
36		bnj966.10	. . . . . . . . 9 ⊢ 𝐷 = (ω ∖ {∅})
37	36	bnj923 34075	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
38	30, 37	bnj769 34069	. . . . . . 7 ⊢ (𝜒 → 𝑛 ∈ ω)
39	38	3ad2ant1 1131	. . . . . 6 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑛 ∈ ω)
40		simp3 1136	. . . . . 6 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖) → 𝑛 = suc 𝑖)
41	39, 40	bnj240 34006	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝑛 ∈ ω ∧ 𝑛 = suc 𝑖))
42		vex 3476	. . . . . . 7 ⊢ 𝑖 ∈ V
43	42	bnj216 34039	. . . . . 6 ⊢ (𝑛 = suc 𝑖 → 𝑖 ∈ 𝑛)
44	43	adantl 480	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ 𝑛 = suc 𝑖) → 𝑖 ∈ 𝑛)
45	41, 44	syl 17	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝑖 ∈ 𝑛)
46		bnj658 34058	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑛 = suc 𝑖) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛))
47	46	anim1i 613	. . . . . 6 ⊢ (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑛 = suc 𝑖) ∧ 𝑖 ∈ 𝑛) → ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ∧ 𝑖 ∈ 𝑛))
48		df-bnj17 33994	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ∧ 𝑖 ∈ 𝑛))
49	47, 48	sylibr 233	. . . . 5 ⊢ (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑛 = suc 𝑖) ∧ 𝑖 ∈ 𝑛) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛))
50	8	bnj945 34080	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛) → (𝐺‘𝑖) = (𝑓‘𝑖))
51	49, 50	syl 17	. . . 4 ⊢ (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑛 = suc 𝑖) ∧ 𝑖 ∈ 𝑛) → (𝐺‘𝑖) = (𝑓‘𝑖))
52	35, 45, 51	syl2anc 582	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝐺‘𝑖) = (𝑓‘𝑖))
53	20, 8	bnj958 34247	. . . . 5 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖))
54	53	bnj956 34083	. . . 4 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
55	54	eqeq2d 2741	. . 3 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ((𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
56	52, 55	syl 17	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → ((𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
57	27, 56	mpbird 256	1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ∖ cdif 3946   ∪ cun 3947  ∅c0 4323  {csn 4629  ⟨cop 4635  ∪ ciun 4998  suc csuc 6367  Fun wfun 6538   Fn wfn 6539  ‘cfv 6544  ωcom 7859   ∧ w-bnj17 33993   predc-bnj14 33995   FrSe w-bnj15 33999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7729  ax-reg 9591

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-id 5575  df-eprel 5581  df-fr 5632  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-bnj17 33994

This theorem is referenced by:  bnj910  34255