Theorem bnj1000 35338

Description: Technical lemma for bnj852 35318. 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 3079	. . . . 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 227	. . . 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 3799	. . 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 1943	. . . . . . . 8 ⊢ Ⅎ𝑓 𝑖 ∈ ω
7	6	sbc19.21g 3814	. . . . . . 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 1943	. . . . . . . . . 10 ⊢ Ⅎ𝑓 suc 𝑖 ∈ 𝑁
10	9	sbc19.21g 3814	. . . . . . . . 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 6880	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐺 → (𝑓‘suc 𝑖) = (𝐺‘suc 𝑖))
13		fveq1 6880	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑖) = (𝐺‘𝑖))
14		ax-5 1939	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝑓‘𝑖) → ∀𝑦 𝑤 ∈ (𝑓‘𝑖))
15		bnj1000.16	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
16		nfcv 2924	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦𝑓
17		nfcv 2924	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑦𝑛
18		bnj1000.15	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
19		nfiu1 4991	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑦∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
20	18, 19	nfcxfr 2922	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑦𝐶
21	17, 20	nfop 4853	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑦⟨𝑛, 𝐶⟩
22	21	nfsn 4672	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦{⟨𝑛, 𝐶⟩}
23	16, 22	nfun 4123	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩})
24	15, 23	nfcxfr 2922	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦𝐺
25		nfcv 2924	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦𝑖
26	24, 25	nffv 6891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝐺‘𝑖)
27	26	nfcrii 2919	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐺‘𝑖) → ∀𝑦 𝑤 ∈ (𝐺‘𝑖))
28	14, 27	bnj1316 35217	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑖) = (𝐺‘𝑖) → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
29	13, 28	syl 18	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐺 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
30	12, 29	eqeq12d 2778	. . . . . . . . . 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 6880	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑒 → (𝑓‘suc 𝑖) = (𝑒‘suc 𝑖))
32		fveq1 6880	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑒 → (𝑓‘𝑖) = (𝑒‘𝑖))
33		ax-5 1939	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑖) = (𝑒‘𝑖) → ∀𝑦(𝑓‘𝑖) = (𝑒‘𝑖))
34	33	bnj956 35174	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑖) = (𝑒‘𝑖) → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅))
35	32, 34	syl 18	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑒 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅))
36	31, 35	eqeq12d 2778	. . . . . . . . . 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 6880	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐺 → (𝑒‘suc 𝑖) = (𝐺‘suc 𝑖))
38		fveq1 6880	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝐺 → (𝑒‘𝑖) = (𝐺‘𝑖))
39		ax-5 1939	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝑒‘𝑖) → ∀𝑦 𝑤 ∈ (𝑒‘𝑖))
40	39, 27	bnj1316 35217	. . . . . . . . . . . 12 ⊢ ((𝑒‘𝑖) = (𝐺‘𝑖) → ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
41	38, 40	syl 18	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐺 → ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
42	37, 41	eqeq12d 2778	. . . . . . . . . 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 35145	. . . . . . . . 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 339	. . . . . . . 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 278	. . . . . . 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 339	. . . . . 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 278	. . . . 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 1848	. . . 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 3802	. . . 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 3079	. . . 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 307	. . 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 3799	. . 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 307	. 2 ⊢ ([𝐺 / 𝑓]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
55	1, 54	bitri 278	1 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567   = wceq 1569   ∈ wcel 2142  ∀wral 3078  Vcvv 3454  [wsbc 3743   ∪ cun 3902  {csn 4588  ⟨cop 4594  ∪ ciun 4955  suc csuc 6362  ‘cfv 6536  ωcom 7860   predc-bnj14 35086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-iota 6492  df-fv 6544

This theorem is used by:  bnj965  35339