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Theorem csbfinxpg 34663
 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 34659 . . 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 3890 . 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 3820 . . . . 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 4364 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑁 ∈ ω ↔ 𝐴 / 𝑥𝑁 ∈ ω))
5 sbceq2g 4367 . . . . . . 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 6707 . . . . . . . . 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 34604 . . . . . . . . . . 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 34606 . . . . . . . . . . . . 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 3901 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥ω = ω)
10 csbconstg 3901 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥V = V)
11 csbif 4521 . . . . . . . . . . . . . . 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 3820 . . . . . . . . . . . . . . . . 17 ([𝐴 / 𝑥](𝑛 = 1o𝑧𝑈) ↔ ([𝐴 / 𝑥]𝑛 = 1o[𝐴 / 𝑥]𝑧𝑈))
13 sbcg 3846 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑛 = 1o𝑛 = 1o))
14 sbcel12 4359 . . . . . . . . . . . . . . . . . . 19 ([𝐴 / 𝑥]𝑧𝑈𝐴 / 𝑥𝑧𝐴 / 𝑥𝑈)
15 csbconstg 3901 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
1615eleq1d 2897 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝑈𝑧𝐴 / 𝑥𝑈))
1714, 16syl5bb 285 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑈𝑧𝐴 / 𝑥𝑈))
1813, 17anbi12d 632 . . . . . . . . . . . . . . . . 17 (𝐴𝑉 → (([𝐴 / 𝑥]𝑛 = 1o[𝐴 / 𝑥]𝑧𝑈) ↔ (𝑛 = 1o𝑧𝐴 / 𝑥𝑈)))
1912, 18syl5bb 285 . . . . . . . . . . . . . . . 16 (𝐴𝑉 → ([𝐴 / 𝑥](𝑛 = 1o𝑧𝑈) ↔ (𝑛 = 1o𝑧𝐴 / 𝑥𝑈)))
20 csbconstg 3901 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥∅ = ∅)
21 csbif 4521 . . . . . . . . . . . . . . . . 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 4359 . . . . . . . . . . . . . . . . . . 19 ([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ 𝐴 / 𝑥𝑧𝐴 / 𝑥(V × 𝑈))
23 csbxp 5644 . . . . . . . . . . . . . . . . . . . . 21 𝐴 / 𝑥(V × 𝑈) = (𝐴 / 𝑥V × 𝐴 / 𝑥𝑈)
2410xpeq1d 5578 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝑉 → (𝐴 / 𝑥V × 𝐴 / 𝑥𝑈) = (V × 𝐴 / 𝑥𝑈))
2523, 24syl5eq 2868 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑉𝐴 / 𝑥(V × 𝑈) = (V × 𝐴 / 𝑥𝑈))
2615, 25eleq12d 2907 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥(V × 𝑈) ↔ 𝑧 ∈ (V × 𝐴 / 𝑥𝑈)))
2722, 26syl5bb 285 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ 𝑧 ∈ (V × 𝐴 / 𝑥𝑈)))
28 csbconstg 3901 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥 𝑛, (1st𝑧)⟩ = ⟨ 𝑛, (1st𝑧)⟩)
29 csbconstg 3901 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥𝑛, 𝑧⟩ = ⟨𝑛, 𝑧⟩)
3027, 28, 29ifbieq12d 4493 . . . . . . . . . . . . . . . . 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 2868 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩) = if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))
3219, 20, 31ifbieq12d 4493 . . . . . . . . . . . . . . 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 2868 . . . . . . . . . . . . . 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 7223 . . . . . . . . . . . . 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 2856 . . . . . . . . . . . 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 4814 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 / 𝑥𝑁, 𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝐴 / 𝑥𝑦⟩)
37 csbconstg 3901 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
3837opeq2d 4803 . . . . . . . . . . . . 13 (𝐴𝑉 → ⟨𝐴 / 𝑥𝑁, 𝐴 / 𝑥𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝑦⟩)
3936, 38eqtrd 2856 . . . . . . . . . . . 12 (𝐴𝑉𝐴 / 𝑥𝑁, 𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝑦⟩)
40 rdgeq12 8043 . . . . . . . . . . . 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 586 . . . . . . . . . . 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 2856 . . . . . . . . . 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 6666 . . . . . . . . 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 2868 . . . . . . . 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 2832 . . . . . . 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 281 . . . . . 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 632 . . . . 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 285 . . . 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 2885 . . 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 4388 . . 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 34659 . . 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 2881 . 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 2868 1 (𝐴𝑉𝐴 / 𝑥(𝑈↑↑𝑁) = (𝐴 / 𝑥𝑈↑↑𝐴 / 𝑥𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {cab 2799  Vcvv 3494  [wsbc 3771  ⦋csb 3882  ∅c0 4290  ifcif 4466  ⟨cop 4566  ∪ cuni 4831   × cxp 5547  ‘cfv 6349   ∈ cmpo 7152  ωcom 7574  1st c1st 7681  reccrdg 8039  1oc1o 8089  ↑↑cfinxp 34658 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-xp 5555  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-iota 6308  df-fv 6357  df-oprab 7154  df-mpo 7155  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-finxp 34659 This theorem is referenced by: (None)
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