Theorem csbfinxpg 37389

Description: Distribute proper substitution through Cartesian exponentiation. (Contributed by ML, 25-Oct-2020.)

Assertion

Ref	Expression
csbfinxpg	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑈↑↑𝑁) = (⦋𝐴 / 𝑥⦌𝑈↑↑⦋𝐴 / 𝑥⦌𝑁))

Distinct variable group:   𝑥,𝑁

Allowed substitution hints:   𝐴(𝑥)   𝑈(𝑥)   𝑉(𝑥)

Proof of Theorem csbfinxpg

Dummy variables 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-finxp 37385	. . 3 ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
2	1	csbeq2i 3907	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝑈↑↑𝑁) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
3		sbcan 3838	. . . . 5 ⊢ ([𝐴 / 𝑥](𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)) ↔ ([𝐴 / 𝑥]𝑁 ∈ ω ∧ [𝐴 / 𝑥]∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)))
4		sbcel1g 4416	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑁 ∈ ω ↔ ⦋𝐴 / 𝑥⦌𝑁 ∈ ω))
5		sbceq2g 4419	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) ↔ ∅ = ⦋𝐴 / 𝑥⦌(rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)))
6		csbfv12 6954	. . . . . . . . 9 ⊢ ⦋𝐴 / 𝑥⦌(rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) = (⦋𝐴 / 𝑥⦌rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁)
7		csbrdgg 37330	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩) = rec(⦋𝐴 / 𝑥⦌(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⦋𝐴 / 𝑥⦌⟨𝑁, 𝑦⟩))
8		csbmpo123 37332	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))) = (𝑛 ∈ ⦋𝐴 / 𝑥⦌ω, 𝑧 ∈ ⦋𝐴 / 𝑥⦌V ↦ ⦋𝐴 / 𝑥⦌if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))))
9		csbconstg 3918	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ω = ω)
10		csbconstg 3918	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌V = V)
11		csbif 4583	. . . . . . . . . . . . . . 15 ⊢ ⦋𝐴 / 𝑥⦌if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)) = if([𝐴 / 𝑥](𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ⦋𝐴 / 𝑥⦌∅, ⦋𝐴 / 𝑥⦌if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))
12		sbcan 3838	. . . . . . . . . . . . . . . . 17 ⊢ ([𝐴 / 𝑥](𝑛 = 1o ∧ 𝑧 ∈ 𝑈) ↔ ([𝐴 / 𝑥]𝑛 = 1o ∧ [𝐴 / 𝑥]𝑧 ∈ 𝑈))
13		sbcg 3863	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑛 = 1o ↔ 𝑛 = 1o))
14		sbcel12 4411	. . . . . . . . . . . . . . . . . . 19 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝑈 ↔ ⦋𝐴 / 𝑥⦌𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈)
15		csbconstg 3918	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
16	15	eleq1d 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈 ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈))
17	14, 16	bitrid 283	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ∈ 𝑈 ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈))
18	13, 17	anbi12d 632	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑛 = 1o ∧ [𝐴 / 𝑥]𝑧 ∈ 𝑈) ↔ (𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈)))
19	12, 18	bitrid 283	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑛 = 1o ∧ 𝑧 ∈ 𝑈) ↔ (𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈)))
20		csbconstg 3918	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∅ = ∅)
21		csbif 4583	. . . . . . . . . . . . . . . . 17 ⊢ ⦋𝐴 / 𝑥⦌if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩) = if([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈), ⦋𝐴 / 𝑥⦌⟨∪ 𝑛, (1st ‘𝑧)⟩, ⦋𝐴 / 𝑥⦌⟨𝑛, 𝑧⟩)
22		sbcel12 4411	. . . . . . . . . . . . . . . . . . 19 ⊢ ([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ ⦋𝐴 / 𝑥⦌𝑧 ∈ ⦋𝐴 / 𝑥⦌(V × 𝑈))
23		csbxp 5785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⦋𝐴 / 𝑥⦌(V × 𝑈) = (⦋𝐴 / 𝑥⦌V × ⦋𝐴 / 𝑥⦌𝑈)
24	10	xpeq1d 5714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌V × ⦋𝐴 / 𝑥⦌𝑈) = (V × ⦋𝐴 / 𝑥⦌𝑈))
25	23, 24	eqtrid 2789	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(V × 𝑈) = (V × ⦋𝐴 / 𝑥⦌𝑈))
26	15, 25	eleq12d 2835	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧 ∈ ⦋𝐴 / 𝑥⦌(V × 𝑈) ↔ 𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈)))
27	22, 26	bitrid 283	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ 𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈)))
28		csbconstg 3918	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨∪ 𝑛, (1st ‘𝑧)⟩ = ⟨∪ 𝑛, (1st ‘𝑧)⟩)
29		csbconstg 3918	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝑛, 𝑧⟩ = ⟨𝑛, 𝑧⟩)
30	27, 28, 29	ifbieq12d 4554	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ 𝑉 → if([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈), ⦋𝐴 / 𝑥⦌⟨∪ 𝑛, (1st ‘𝑧)⟩, ⦋𝐴 / 𝑥⦌⟨𝑛, 𝑧⟩) = if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))
31	21, 30	eqtrid 2789	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩) = if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))
32	19, 20, 31	ifbieq12d 4554	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → if([𝐴 / 𝑥](𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ⦋𝐴 / 𝑥⦌∅, ⦋𝐴 / 𝑥⦌if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)) = if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)))
33	11, 32	eqtrid 2789	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)) = if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)))
34	9, 10, 33	mpoeq123dv 7508	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (𝑛 ∈ ⦋𝐴 / 𝑥⦌ω, 𝑧 ∈ ⦋𝐴 / 𝑥⦌V ↦ ⦋𝐴 / 𝑥⦌if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))) = (𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))))
35	8, 34	eqtrd 2777	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))) = (𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))))
36		csbopg 4891	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝑁, 𝑦⟩ = ⟨⦋𝐴 / 𝑥⦌𝑁, ⦋𝐴 / 𝑥⦌𝑦⟩)
37		csbconstg 3918	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
38	37	opeq2d 4880	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ⟨⦋𝐴 / 𝑥⦌𝑁, ⦋𝐴 / 𝑥⦌𝑦⟩ = ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)
39	36, 38	eqtrd 2777	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝑁, 𝑦⟩ = ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)
40		rdgeq12 8453	. . . . . . . . . . . 12 ⊢ ((⦋𝐴 / 𝑥⦌(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))) = (𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))) ∧ ⦋𝐴 / 𝑥⦌⟨𝑁, 𝑦⟩ = ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩) → rec(⦋𝐴 / 𝑥⦌(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⦋𝐴 / 𝑥⦌⟨𝑁, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩))
41	35, 39, 40	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → rec(⦋𝐴 / 𝑥⦌(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⦋𝐴 / 𝑥⦌⟨𝑁, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩))
42	7, 41	eqtrd 2777	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩))
43	42	fveq1d 6908	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁) = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁))
44	6, 43	eqtrid 2789	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁))
45	44	eqeq2d 2748	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (∅ = ⦋𝐴 / 𝑥⦌(rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁)))
46	5, 45	bitrd 279	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁)))
47	4, 46	anbi12d 632	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑁 ∈ ω ∧ [𝐴 / 𝑥]∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)) ↔ (⦋𝐴 / 𝑥⦌𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁))))
48	3, 47	bitrid 283	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)) ↔ (⦋𝐴 / 𝑥⦌𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁))))
49	48	abbidv 2808	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥](𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁))})
50		csbab 4440	. . 3 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} = {𝑦 ∣ [𝐴 / 𝑥](𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
51		df-finxp 37385	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝑈↑↑⦋𝐴 / 𝑥⦌𝑁) = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁))}
52	49, 50, 51	3eqtr4g 2802	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} = (⦋𝐴 / 𝑥⦌𝑈↑↑⦋𝐴 / 𝑥⦌𝑁))
53	2, 52	eqtrid 2789	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑈↑↑𝑁) = (⦋𝐴 / 𝑥⦌𝑈↑↑⦋𝐴 / 𝑥⦌𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  Vcvv 3480  [wsbc 3788  ⦋csb 3899  ∅c0 4333  ifcif 4525  ⟨cop 4632  ∪ cuni 4907   × cxp 5683  ‘cfv 6561   ∈ cmpo 7433  ωcom 7887  1^st c1st 8012  reccrdg 8449  1_oc1o 8499  ↑↑cfinxp 37384

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-finxp 37385

This theorem is referenced by: (None)