Theorem cbvprod 11534

Description: Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Hypotheses

Ref	Expression
cbvprod.1	⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)
cbvprod.2	⊢ Ⅎ𝑘𝐴
cbvprod.3	⊢ Ⅎ𝑗𝐴
cbvprod.4	⊢ Ⅎ𝑘𝐵
cbvprod.5	⊢ Ⅎ𝑗𝐶

Assertion

Ref	Expression
cbvprod	⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶

Distinct variable group:   𝑗,𝑘

Allowed substitution hints:   𝐴(𝑗,𝑘)   𝐵(𝑗,𝑘)   𝐶(𝑗,𝑘)

Proof of Theorem cbvprod

Dummy variables 𝑓 𝑚 𝑛 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvprod.2	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐴
2	1	nfcri 2311	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑗 ∈ 𝐴
3		cbvprod.4	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘𝐵
4		nfcv 2317	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘1
5	2, 3, 4	nfif 3560	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘if(𝑗 ∈ 𝐴, 𝐵, 1)
6		cbvprod.3	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝐴
7	6	nfcri 2311	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗 𝑘 ∈ 𝐴
8		cbvprod.5	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝐶
9		nfcv 2317	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗1
10	7, 8, 9	nfif 3560	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗if(𝑘 ∈ 𝐴, 𝐶, 1)
11		eleq1w 2236	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
12		cbvprod.1	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)
13	11, 12	ifbieq1d 3554	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → if(𝑗 ∈ 𝐴, 𝐵, 1) = if(𝑘 ∈ 𝐴, 𝐶, 1))
14	5, 10, 13	cbvmpt 4093	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))
15		seqeq3 10420	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1)) → seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))))
16	14, 15	ax-mp 5	. . . . . . . . . . 11 ⊢ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1)))
17	16	breq1i 4005	. . . . . . . . . 10 ⊢ (seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦)
18	17	anbi2i 457	. . . . . . . . 9 ⊢ ((𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ (𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦))
19	18	exbii 1603	. . . . . . . 8 ⊢ (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦))
20	19	rexbii 2482	. . . . . . 7 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦))
21		seqeq3 10420	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1)) → seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))))
22	14, 21	ax-mp 5	. . . . . . . 8 ⊢ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1)))
23	22	breq1i 4005	. . . . . . 7 ⊢ (seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥)
24	20, 23	anbi12i 460	. . . . . 6 ⊢ ((∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥))
25	24	anbi2i 457	. . . . 5 ⊢ (((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑠 ∈ (ℤ≥‘𝑚)DECID 𝑠 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ↔ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑠 ∈ (ℤ≥‘𝑚)DECID 𝑠 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥)))
26	25	rexbii 2482	. . . 4 ⊢ (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑠 ∈ (ℤ≥‘𝑚)DECID 𝑠 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ↔ ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑠 ∈ (ℤ≥‘𝑚)DECID 𝑠 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥)))
27	3, 8, 12	cbvcsbw 3059	. . . . . . . . . . . 12 ⊢ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶
28		ifeq1 3535	. . . . . . . . . . . 12 ⊢ (⦋(𝑓‘𝑛) / 𝑗⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶 → if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1) = if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1))
29	27, 28	ax-mp 5	. . . . . . . . . . 11 ⊢ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1) = if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)
30	29	mpteq2i 4085	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1))
31		seqeq3 10420	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)) → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1))) = seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1))))
32	30, 31	ax-mp 5	. . . . . . . . 9 ⊢ seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1))) = seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))
33	32	fveq1i 5508	. . . . . . . 8 ⊢ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1)))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚)
34	33	eqeq2i 2186	. . . . . . 7 ⊢ (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1)))‘𝑚) ↔ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚))
35	34	anbi2i 457	. . . . . 6 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1)))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚)))
36	35	exbii 1603	. . . . 5 ⊢ (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚)))
37	36	rexbii 2482	. . . 4 ⊢ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1)))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚)))
38	26, 37	orbi12i 764	. . 3 ⊢ ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑠 ∈ (ℤ≥‘𝑚)DECID 𝑠 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1)))‘𝑚))) ↔ (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑠 ∈ (ℤ≥‘𝑚)DECID 𝑠 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚))))
39	38	iotabii 5192	. 2 ⊢ (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑠 ∈ (ℤ≥‘𝑚)DECID 𝑠 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1)))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑠 ∈ (ℤ≥‘𝑚)DECID 𝑠 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚))))
40		df-proddc 11527	. 2 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑠 ∈ (ℤ≥‘𝑚)DECID 𝑠 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑗⦌𝐵, 1)))‘𝑚))))
41		df-proddc 11527	. 2 ⊢ ∏𝑘 ∈ 𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑠 ∈ (ℤ≥‘𝑚)DECID 𝑠 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚))))
42	39, 40, 41	3eqtr4i 2206	1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 708  DECID wdc 834   = wceq 1353  ∃wex 1490   ∈ wcel 2146  Ⅎwnfc 2304  ∀wral 2453  ∃wrex 2454  ⦋csb 3055   ⊆ wss 3127  ifcif 3532   class class class wbr 3998   ↦ cmpt 4059  ℩cio 5168  –1-1-onto→wf1o 5207  ‘cfv 5208  (class class class)co 5865  0cc0 7786  1c1 7787   · cmul 7791   ≤ cle 7967   # cap 8512  ℕcn 8892  ℤcz 9226  ℤ_≥cuz 9501  ...cfz 9979  seqcseq 10415   ⇝ cli 11254  ∏cprod 11526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-if 3533  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-cnv 4628  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-recs 6296  df-frec 6382  df-seqfrec 10416  df-proddc 11527

This theorem is referenced by:  cbvprodv  11535  cbvprodi  11536