Theorem bnj1000 30265

Description: Technical lemma for bnj852 30245. 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
bnj1000.1	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1000.2	⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓)
bnj1000.3	⊢ 𝐺 ∈ V
bnj1000.15	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj1000.16	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})

Assertion

Ref	Expression
bnj1000	⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Distinct variable groups:   𝐴,𝑓   𝑖,𝐺   𝑓,𝑁   𝑅,𝑓   𝑓,𝑖,𝑦   𝑦,𝑛

Allowed substitution hints:   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛)   𝐴(𝑦,𝑖,𝑚,𝑛)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛)   𝑅(𝑦,𝑖,𝑚,𝑛)   𝐺(𝑦,𝑓,𝑚,𝑛)   𝑁(𝑦,𝑖,𝑚,𝑛)   𝜓″(𝑦,𝑓,𝑖,𝑚,𝑛)

Proof of Theorem bnj1000

Dummy variables 𝑒 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1000.2	. 2 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓)
2		df-ral 2901	. . . . 5 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
3	2	bicomi 213	. . . 4 ⊢ (∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
4	3	sbcbii 3458	. . 3 ⊢ ([𝐺 / 𝑓]∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ [𝐺 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
5		bnj1000.3	. . . . . . 7 ⊢ 𝐺 ∈ V
6		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑓 𝑖 ∈ ω
7	6	sbc19.21g 3469	. . . . . . 7 ⊢ (𝐺 ∈ V → ([𝐺 / 𝑓](𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑖 ∈ ω → [𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))))
8	5, 7	ax-mp 5	. . . . . 6 ⊢ ([𝐺 / 𝑓](𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑖 ∈ ω → [𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
9		nfv 1830	. . . . . . . . . 10 ⊢ Ⅎ𝑓 suc 𝑖 ∈ 𝑁
10	9	sbc19.21g 3469	. . . . . . . . 9 ⊢ (𝐺 ∈ V → ([𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑁 → [𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
11	5, 10	ax-mp 5	. . . . . . . 8 ⊢ ([𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑁 → [𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
12		fveq1 6102	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐺 → (𝑓‘suc 𝑖) = (𝐺‘suc 𝑖))
13		fveq1 6102	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑖) = (𝐺‘𝑖))
14		ax-5 1827	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝑓‘𝑖) → ∀𝑦 𝑤 ∈ (𝑓‘𝑖))
15		bnj1000.16	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
16		nfcv 2751	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦𝑓
17		nfcv 2751	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑦𝑛
18		bnj1000.15	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
19		nfiu1 4486	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑦∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
20	18, 19	nfcxfr 2749	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑦𝐶
21	17, 20	nfop 4356	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑦⟨𝑛, 𝐶⟩
22	21	nfsn 4189	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦{⟨𝑛, 𝐶⟩}
23	16, 22	nfun 3731	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩})
24	15, 23	nfcxfr 2749	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦𝐺
25		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦𝑖
26	24, 25	nffv 6110	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝐺‘𝑖)
27	26	nfcrii 2744	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐺‘𝑖) → ∀𝑦 𝑤 ∈ (𝐺‘𝑖))
28	14, 27	bnj1316 30145	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑖) = (𝐺‘𝑖) → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
29	13, 28	syl 17	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐺 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
30	12, 29	eqeq12d 2625	. . . . . . . . . 10 ⊢ (𝑓 = 𝐺 → ((𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
31		fveq1 6102	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑒 → (𝑓‘suc 𝑖) = (𝑒‘suc 𝑖))
32		fveq1 6102	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑒 → (𝑓‘𝑖) = (𝑒‘𝑖))
33		ax-5 1827	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑖) = (𝑒‘𝑖) → ∀𝑦(𝑓‘𝑖) = (𝑒‘𝑖))
34	33	bnj956 30101	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑖) = (𝑒‘𝑖) → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅))
35	32, 34	syl 17	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑒 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅))
36	31, 35	eqeq12d 2625	. . . . . . . . . 10 ⊢ (𝑓 = 𝑒 → ((𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑒‘suc 𝑖) = ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅)))
37		fveq1 6102	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐺 → (𝑒‘suc 𝑖) = (𝐺‘suc 𝑖))
38		fveq1 6102	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝐺 → (𝑒‘𝑖) = (𝐺‘𝑖))
39		ax-5 1827	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝑒‘𝑖) → ∀𝑦 𝑤 ∈ (𝑒‘𝑖))
40	39, 27	bnj1316 30145	. . . . . . . . . . . 12 ⊢ ((𝑒‘𝑖) = (𝐺‘𝑖) → ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
41	38, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐺 → ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
42	37, 41	eqeq12d 2625	. . . . . . . . . 10 ⊢ (𝑒 = 𝐺 → ((𝑒‘suc 𝑖) = ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
43	5, 30, 36, 42	bnj610 30071	. . . . . . . . 9 ⊢ ([𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
44	43	imbi2i 325	. . . . . . . 8 ⊢ ((suc 𝑖 ∈ 𝑁 → [𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
45	11, 44	bitri 263	. . . . . . 7 ⊢ ([𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
46	45	imbi2i 325	. . . . . 6 ⊢ ((𝑖 ∈ ω → [𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))))
47	8, 46	bitri 263	. . . . 5 ⊢ ([𝐺 / 𝑓](𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))))
48	47	albii 1737	. . . 4 ⊢ (∀𝑖[𝐺 / 𝑓](𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))))
49		sbcal 3452	. . . 4 ⊢ ([𝐺 / 𝑓]∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ∀𝑖[𝐺 / 𝑓](𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
50		df-ral 2901	. . . 4 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))))
51	48, 49, 50	3bitr4ri 292	. . 3 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [𝐺 / 𝑓]∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
52		bnj1000.1	. . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
53	52	sbcbii 3458	. . 3 ⊢ ([𝐺 / 𝑓]𝜓 ↔ [𝐺 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
54	4, 51, 53	3bitr4ri 292	. 2 ⊢ ([𝐺 / 𝑓]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
55	1, 54	bitri 263	1 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402   ∪ cun 3538  {csn 4125  ⟨cop 4131  ∪ ciun 4455  suc csuc 5642  ‘cfv 5804  ωcom 6957   predc-bnj14 30007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-iota 5768  df-fv 5812

This theorem is referenced by:  bnj965  30266