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Theorem csbfinxpg 32401
 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 32397 . . 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 3945 . 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 3445 . . . . 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 3939 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑁 ∈ ω ↔ 𝐴 / 𝑥𝑁 ∈ ω))
5 sbceq2g 3942 . . . . . . 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 6141 . . . . . . . . 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 32351 . . . . . . . . . . 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 32353 . . . . . . . . . . . . 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 3512 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥ω = ω)
10 csbconstg 3512 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥V = V)
11 csbif 4088 . . . . . . . . . . . . . . 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 3445 . . . . . . . . . . . . . . . . 17 ([𝐴 / 𝑥](𝑛 = 1𝑜𝑧𝑈) ↔ ([𝐴 / 𝑥]𝑛 = 1𝑜[𝐴 / 𝑥]𝑧𝑈))
13 sbcg 3470 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑛 = 1𝑜𝑛 = 1𝑜))
14 sbcel12 3935 . . . . . . . . . . . . . . . . . . 19 ([𝐴 / 𝑥]𝑧𝑈𝐴 / 𝑥𝑧𝐴 / 𝑥𝑈)
15 csbconstg 3512 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
1615eleq1d 2672 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝑈𝑧𝐴 / 𝑥𝑈))
1714, 16syl5bb 271 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑈𝑧𝐴 / 𝑥𝑈))
1813, 17anbi12d 743 . . . . . . . . . . . . . . . . 17 (𝐴𝑉 → (([𝐴 / 𝑥]𝑛 = 1𝑜[𝐴 / 𝑥]𝑧𝑈) ↔ (𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈)))
1912, 18syl5bb 271 . . . . . . . . . . . . . . . 16 (𝐴𝑉 → ([𝐴 / 𝑥](𝑛 = 1𝑜𝑧𝑈) ↔ (𝑛 = 1𝑜𝑧𝐴 / 𝑥𝑈)))
20 csbconstg 3512 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥∅ = ∅)
21 csbif 4088 . . . . . . . . . . . . . . . . 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 3935 . . . . . . . . . . . . . . . . . . 19 ([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ 𝐴 / 𝑥𝑧𝐴 / 𝑥(V × 𝑈))
23 csbxp 5123 . . . . . . . . . . . . . . . . . . . . 21 𝐴 / 𝑥(V × 𝑈) = (𝐴 / 𝑥V × 𝐴 / 𝑥𝑈)
2410xpeq1d 5062 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝑉 → (𝐴 / 𝑥V × 𝐴 / 𝑥𝑈) = (V × 𝐴 / 𝑥𝑈))
2523, 24syl5eq 2656 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑉𝐴 / 𝑥(V × 𝑈) = (V × 𝐴 / 𝑥𝑈))
2615, 25eleq12d 2682 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥(V × 𝑈) ↔ 𝑧 ∈ (V × 𝐴 / 𝑥𝑈)))
2722, 26syl5bb 271 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ 𝑧 ∈ (V × 𝐴 / 𝑥𝑈)))
28 csbconstg 3512 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥 𝑛, (1st𝑧)⟩ = ⟨ 𝑛, (1st𝑧)⟩)
29 csbconstg 3512 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥𝑛, 𝑧⟩ = ⟨𝑛, 𝑧⟩)
3027, 28, 29ifbieq12d 4063 . . . . . . . . . . . . . . . . 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 2656 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩) = if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))
3219, 20, 31ifbieq12d 4063 . . . . . . . . . . . . . . 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 2656 . . . . . . . . . . . . . 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 6615 . . . . . . . . . . . . 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 2644 . . . . . . . . . . . 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 4358 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 / 𝑥𝑁, 𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝐴 / 𝑥𝑦⟩)
37 csbconstg 3512 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
3837opeq2d 4347 . . . . . . . . . . . . 13 (𝐴𝑉 → ⟨𝐴 / 𝑥𝑁, 𝐴 / 𝑥𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝑦⟩)
3936, 38eqtrd 2644 . . . . . . . . . . . 12 (𝐴𝑉𝐴 / 𝑥𝑁, 𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝑦⟩)
40 rdgeq12 7396 . . . . . . . . . . . 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 691 . . . . . . . . . . 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 2644 . . . . . . . . . 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 6105 . . . . . . . . 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 2656 . . . . . . . 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 2620 . . . . . . 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 267 . . . . . 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 743 . . . . 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 271 . . . 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 2728 . . 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 3960 . . 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 32397 . . 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 2669 . 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 2656 1 (𝐴𝑉𝐴 / 𝑥(𝑈↑↑𝑁) = (𝐴 / 𝑥𝑈↑↑𝐴 / 𝑥𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173  [wsbc 3402  ⦋csb 3499  ∅c0 3874  ifcif 4036  ⟨cop 4131  ∪ cuni 4372   × cxp 5036  ‘cfv 5804   ↦ cmpt2 6551  ωcom 6957  1st c1st 7057  reccrdg 7392  1𝑜c1o 7440  ↑↑cfinxp 32396 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-finxp 32397 This theorem is referenced by: (None)
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