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Theorem csbfinxpg 35797
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 35793 . . 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 3861 . 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 3789 . . . . 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 4371 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑁 ∈ ω ↔ 𝐴 / 𝑥𝑁 ∈ ω))
5 sbceq2g 4374 . . . . . . 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 6887 . . . . . . . . 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 35738 . . . . . . . . . . 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 csbmpo123 35740 . . . . . . . . . . . . 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 3872 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥ω = ω)
10 csbconstg 3872 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥V = V)
11 csbif 4541 . . . . . . . . . . . . . . 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 3789 . . . . . . . . . . . . . . . . 17 ([𝐴 / 𝑥](𝑛 = 1o𝑧𝑈) ↔ ([𝐴 / 𝑥]𝑛 = 1o[𝐴 / 𝑥]𝑧𝑈))
13 sbcg 3816 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑛 = 1o𝑛 = 1o))
14 sbcel12 4366 . . . . . . . . . . . . . . . . . . 19 ([𝐴 / 𝑥]𝑧𝑈𝐴 / 𝑥𝑧𝐴 / 𝑥𝑈)
15 csbconstg 3872 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
1615eleq1d 2822 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝑈𝑧𝐴 / 𝑥𝑈))
1714, 16bitrid 282 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑈𝑧𝐴 / 𝑥𝑈))
1813, 17anbi12d 631 . . . . . . . . . . . . . . . . 17 (𝐴𝑉 → (([𝐴 / 𝑥]𝑛 = 1o[𝐴 / 𝑥]𝑧𝑈) ↔ (𝑛 = 1o𝑧𝐴 / 𝑥𝑈)))
1912, 18bitrid 282 . . . . . . . . . . . . . . . 16 (𝐴𝑉 → ([𝐴 / 𝑥](𝑛 = 1o𝑧𝑈) ↔ (𝑛 = 1o𝑧𝐴 / 𝑥𝑈)))
20 csbconstg 3872 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥∅ = ∅)
21 csbif 4541 . . . . . . . . . . . . . . . . 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 4366 . . . . . . . . . . . . . . . . . . 19 ([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ 𝐴 / 𝑥𝑧𝐴 / 𝑥(V × 𝑈))
23 csbxp 5729 . . . . . . . . . . . . . . . . . . . . 21 𝐴 / 𝑥(V × 𝑈) = (𝐴 / 𝑥V × 𝐴 / 𝑥𝑈)
2410xpeq1d 5660 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝑉 → (𝐴 / 𝑥V × 𝐴 / 𝑥𝑈) = (V × 𝐴 / 𝑥𝑈))
2523, 24eqtrid 2789 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑉𝐴 / 𝑥(V × 𝑈) = (V × 𝐴 / 𝑥𝑈))
2615, 25eleq12d 2832 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥(V × 𝑈) ↔ 𝑧 ∈ (V × 𝐴 / 𝑥𝑈)))
2722, 26bitrid 282 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ 𝑧 ∈ (V × 𝐴 / 𝑥𝑈)))
28 csbconstg 3872 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥 𝑛, (1st𝑧)⟩ = ⟨ 𝑛, (1st𝑧)⟩)
29 csbconstg 3872 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥𝑛, 𝑧⟩ = ⟨𝑛, 𝑧⟩)
3027, 28, 29ifbieq12d 4512 . . . . . . . . . . . . . . . . 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, 30eqtrid 2789 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩) = if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))
3219, 20, 31ifbieq12d 4512 . . . . . . . . . . . . . . 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, 32eqtrid 2789 . . . . . . . . . . . . . 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, 33mpoeq123dv 7426 . . . . . . . . . . . . 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 2777 . . . . . . . . . . . 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 4846 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 / 𝑥𝑁, 𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝐴 / 𝑥𝑦⟩)
37 csbconstg 3872 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
3837opeq2d 4835 . . . . . . . . . . . . 13 (𝐴𝑉 → ⟨𝐴 / 𝑥𝑁, 𝐴 / 𝑥𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝑦⟩)
3936, 38eqtrd 2777 . . . . . . . . . . . 12 (𝐴𝑉𝐴 / 𝑥𝑁, 𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝑦⟩)
40 rdgeq12 8351 . . . . . . . . . . . 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 584 . . . . . . . . . . 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 2777 . . . . . . . . . 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 6841 . . . . . . . . 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, 43eqtrid 2789 . . . . . . . 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 2748 . . . . . . 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 278 . . . . . 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 631 . . . . 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, 47bitrid 282 . . . 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 2806 . . 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 4395 . . 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 35793 . . 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 2802 . 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, 52eqtrid 2789 1 (𝐴𝑉𝐴 / 𝑥(𝑈↑↑𝑁) = (𝐴 / 𝑥𝑈↑↑𝐴 / 𝑥𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {cab 2714  Vcvv 3443  [wsbc 3737  csb 3853  c0 4280  ifcif 4484  cop 4590   cuni 4863   × cxp 5629  cfv 6493  cmpo 7353  ωcom 7794  1st c1st 7911  reccrdg 8347  1oc1o 8397  ↑↑cfinxp 35792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-xp 5637  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-iota 6445  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-finxp 35793
This theorem is referenced by: (None)
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