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Theorem csbfinxpg 33554
Description: Distribute proper substitution through Cartesian exponentiation. (Contributed by ML, 25-Oct-2020.)
Assertion
Ref Expression
csbfinxpg (𝐴𝑉𝐴 / 𝑥(𝑈↑↑𝑁) = (𝐴 / 𝑥𝑈↑↑𝐴 / 𝑥𝑁))
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.
StepHypRef Expression
1 df-finxp 33550 . . 3 (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
21csbeq2i 4136 . 2 𝐴 / 𝑥(𝑈↑↑𝑁) = 𝐴 / 𝑥{𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
3 sbcan 3619 . . . . 5 ([𝐴 / 𝑥](𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)) ↔ ([𝐴 / 𝑥]𝑁 ∈ ω ∧ [𝐴 / 𝑥]∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)))
4 sbcel1g 4130 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑁 ∈ ω ↔ 𝐴 / 𝑥𝑁 ∈ ω))
5 sbceq2g 4133 . . . . . . 7 (𝐴𝑉 → ([𝐴 / 𝑥]∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) ↔ ∅ = 𝐴 / 𝑥(rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)))
6 csbfv12 6393 . . . . . . . . 9 𝐴 / 𝑥(rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) = (𝐴 / 𝑥rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝐴 / 𝑥𝑁)
7 csbrdgg 33504 . . . . . . . . . . 11 (𝐴𝑉𝐴 / 𝑥rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩) = rec(𝐴 / 𝑥(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), 𝐴 / 𝑥𝑁, 𝑦⟩))
8 csbmpt22g 33506 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 / 𝑥(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))) = (𝑛𝐴 / 𝑥ω, 𝑧𝐴 / 𝑥V ↦ 𝐴 / 𝑥if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))))
9 csbconstg 3687 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥ω = ω)
10 csbconstg 3687 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥V = V)
11 csbif 4282 . . . . . . . . . . . . . . 15 𝐴 / 𝑥if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩)) = if([𝐴 / 𝑥](𝑛 = 1𝑜𝑧𝑈), 𝐴 / 𝑥∅, 𝐴 / 𝑥if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))
12 sbcan 3619 . . . . . . . . . . . . . . . . 17 ([𝐴 / 𝑥](𝑛 = 1𝑜𝑧𝑈) ↔ ([𝐴 / 𝑥]𝑛 = 1𝑜[𝐴 / 𝑥]𝑧𝑈))
13 sbcg 3644 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑛 = 1𝑜𝑛 = 1𝑜))
14 sbcel12 4126 . . . . . . . . . . . . . . . . . . 19 ([𝐴 / 𝑥]𝑧𝑈𝐴 / 𝑥𝑧𝐴 / 𝑥𝑈)
15 csbconstg 3687 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
1615eleq1d 2824 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝑈𝑧𝐴 / 𝑥𝑈))
1714, 16syl5bb 272 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑈𝑧𝐴 / 𝑥𝑈))
1813, 17anbi12d 749 . . . . . . . . . . . . . . . . 17 (𝐴𝑉 → (([𝐴 / 𝑥]𝑛 = 1𝑜[𝐴 / 𝑥]𝑧𝑈) ↔ (𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈)))
1912, 18syl5bb 272 . . . . . . . . . . . . . . . 16 (𝐴𝑉 → ([𝐴 / 𝑥](𝑛 = 1𝑜𝑧𝑈) ↔ (𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈)))
20 csbconstg 3687 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥∅ = ∅)
21 csbif 4282 . . . . . . . . . . . . . . . . 17 𝐴 / 𝑥if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩) = if([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈), 𝐴 / 𝑥 𝑛, (1st𝑧)⟩, 𝐴 / 𝑥𝑛, 𝑧⟩)
22 sbcel12 4126 . . . . . . . . . . . . . . . . . . 19 ([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ 𝐴 / 𝑥𝑧𝐴 / 𝑥(V × 𝑈))
23 csbxp 5357 . . . . . . . . . . . . . . . . . . . . 21 𝐴 / 𝑥(V × 𝑈) = (𝐴 / 𝑥V × 𝐴 / 𝑥𝑈)
2410xpeq1d 5295 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝑉 → (𝐴 / 𝑥V × 𝐴 / 𝑥𝑈) = (V × 𝐴 / 𝑥𝑈))
2523, 24syl5eq 2806 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑉𝐴 / 𝑥(V × 𝑈) = (V × 𝐴 / 𝑥𝑈))
2615, 25eleq12d 2833 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥(V × 𝑈) ↔ 𝑧 ∈ (V × 𝐴 / 𝑥𝑈)))
2722, 26syl5bb 272 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ 𝑧 ∈ (V × 𝐴 / 𝑥𝑈)))
28 csbconstg 3687 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥 𝑛, (1st𝑧)⟩ = ⟨ 𝑛, (1st𝑧)⟩)
29 csbconstg 3687 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥𝑛, 𝑧⟩ = ⟨𝑛, 𝑧⟩)
3027, 28, 29ifbieq12d 4257 . . . . . . . . . . . . . . . . 17 (𝐴𝑉 → if([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈), 𝐴 / 𝑥 𝑛, (1st𝑧)⟩, 𝐴 / 𝑥𝑛, 𝑧⟩) = if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))
3121, 30syl5eq 2806 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩) = if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))
3219, 20, 31ifbieq12d 4257 . . . . . . . . . . . . . . 15 (𝐴𝑉 → if([𝐴 / 𝑥](𝑛 = 1𝑜𝑧𝑈), 𝐴 / 𝑥∅, 𝐴 / 𝑥if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩)) = if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩)))
3311, 32syl5eq 2806 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩)) = if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩)))
349, 10, 33mpt2eq123dv 6883 . . . . . . . . . . . . 13 (𝐴𝑉 → (𝑛𝐴 / 𝑥ω, 𝑧𝐴 / 𝑥V ↦ 𝐴 / 𝑥if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))) = (𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))))
358, 34eqtrd 2794 . . . . . . . . . . . 12 (𝐴𝑉𝐴 / 𝑥(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))) = (𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))))
36 csbopg 4571 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 / 𝑥𝑁, 𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝐴 / 𝑥𝑦⟩)
37 csbconstg 3687 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
3837opeq2d 4560 . . . . . . . . . . . . 13 (𝐴𝑉 → ⟨𝐴 / 𝑥𝑁, 𝐴 / 𝑥𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝑦⟩)
3936, 38eqtrd 2794 . . . . . . . . . . . 12 (𝐴𝑉𝐴 / 𝑥𝑁, 𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝑦⟩)
40 rdgeq12 7679 . . . . . . . . . . . 12 ((𝐴 / 𝑥(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))) = (𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))) ∧ 𝐴 / 𝑥𝑁, 𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝑦⟩) → rec(𝐴 / 𝑥(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), 𝐴 / 𝑥𝑁, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝐴 / 𝑥𝑁, 𝑦⟩))
4135, 39, 40syl2anc 696 . . . . . . . . . . 11 (𝐴𝑉 → rec(𝐴 / 𝑥(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), 𝐴 / 𝑥𝑁, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝐴 / 𝑥𝑁, 𝑦⟩))
427, 41eqtrd 2794 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝐴 / 𝑥𝑁, 𝑦⟩))
4342fveq1d 6355 . . . . . . . . 9 (𝐴𝑉 → (𝐴 / 𝑥rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝐴 / 𝑥𝑁) = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝐴 / 𝑥𝑁, 𝑦⟩)‘𝐴 / 𝑥𝑁))
446, 43syl5eq 2806 . . . . . . . 8 (𝐴𝑉𝐴 / 𝑥(rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝐴 / 𝑥𝑁, 𝑦⟩)‘𝐴 / 𝑥𝑁))
4544eqeq2d 2770 . . . . . . 7 (𝐴𝑉 → (∅ = 𝐴 / 𝑥(rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝐴 / 𝑥𝑁, 𝑦⟩)‘𝐴 / 𝑥𝑁)))
465, 45bitrd 268 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝐴 / 𝑥𝑁, 𝑦⟩)‘𝐴 / 𝑥𝑁)))
474, 46anbi12d 749 . . . . 5 (𝐴𝑉 → (([𝐴 / 𝑥]𝑁 ∈ ω ∧ [𝐴 / 𝑥]∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)) ↔ (𝐴 / 𝑥𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝐴 / 𝑥𝑁, 𝑦⟩)‘𝐴 / 𝑥𝑁))))
483, 47syl5bb 272 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)) ↔ (𝐴 / 𝑥𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝐴 / 𝑥𝑁, 𝑦⟩)‘𝐴 / 𝑥𝑁))))
4948abbidv 2879 . . 3 (𝐴𝑉 → {𝑦[𝐴 / 𝑥](𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} = {𝑦 ∣ (𝐴 / 𝑥𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝐴 / 𝑥𝑁, 𝑦⟩)‘𝐴 / 𝑥𝑁))})
50 csbab 4151 . . 3 𝐴 / 𝑥{𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} = {𝑦[𝐴 / 𝑥](𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
51 df-finxp 33550 . . 3 (𝐴 / 𝑥𝑈↑↑𝐴 / 𝑥𝑁) = {𝑦 ∣ (𝐴 / 𝑥𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈), ∅, if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝐴 / 𝑥𝑁, 𝑦⟩)‘𝐴 / 𝑥𝑁))}
5249, 50, 513eqtr4g 2819 . 2 (𝐴𝑉𝐴 / 𝑥{𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1𝑜𝑧𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} = (𝐴 / 𝑥𝑈↑↑𝐴 / 𝑥𝑁))
532, 52syl5eq 2806 1 (𝐴𝑉𝐴 / 𝑥(𝑈↑↑𝑁) = (𝐴 / 𝑥𝑈↑↑𝐴 / 𝑥𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {cab 2746  Vcvv 3340  [wsbc 3576  csb 3674  c0 4058  ifcif 4230  cop 4327   cuni 4588   × cxp 5264  cfv 6049  cmpt2 6816  ωcom 7231  1st c1st 7332  reccrdg 7675  1𝑜c1o 7723  ↑↑cfinxp 33549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-iota 6012  df-fv 6057  df-oprab 6818  df-mpt2 6819  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-finxp 33550
This theorem is referenced by: (None)
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