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Theorem csbfinxpg 37371
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 37367 . . 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 3916 . 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 3844 . . . . 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 4422 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑁 ∈ ω ↔ 𝐴 / 𝑥𝑁 ∈ ω))
5 sbceq2g 4425 . . . . . . 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 6955 . . . . . . . . 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 37312 . . . . . . . . . . 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 37314 . . . . . . . . . . . . 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 3927 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥ω = ω)
10 csbconstg 3927 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥V = V)
11 csbif 4588 . . . . . . . . . . . . . . 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 3844 . . . . . . . . . . . . . . . . 17 ([𝐴 / 𝑥](𝑛 = 1o𝑧𝑈) ↔ ([𝐴 / 𝑥]𝑛 = 1o[𝐴 / 𝑥]𝑧𝑈))
13 sbcg 3870 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑛 = 1o𝑛 = 1o))
14 sbcel12 4417 . . . . . . . . . . . . . . . . . . 19 ([𝐴 / 𝑥]𝑧𝑈𝐴 / 𝑥𝑧𝐴 / 𝑥𝑈)
15 csbconstg 3927 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
1615eleq1d 2824 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝑈𝑧𝐴 / 𝑥𝑈))
1714, 16bitrid 283 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑈𝑧𝐴 / 𝑥𝑈))
1813, 17anbi12d 632 . . . . . . . . . . . . . . . . 17 (𝐴𝑉 → (([𝐴 / 𝑥]𝑛 = 1o[𝐴 / 𝑥]𝑧𝑈) ↔ (𝑛 = 1o𝑧𝐴 / 𝑥𝑈)))
1912, 18bitrid 283 . . . . . . . . . . . . . . . 16 (𝐴𝑉 → ([𝐴 / 𝑥](𝑛 = 1o𝑧𝑈) ↔ (𝑛 = 1o𝑧𝐴 / 𝑥𝑈)))
20 csbconstg 3927 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥∅ = ∅)
21 csbif 4588 . . . . . . . . . . . . . . . . 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 4417 . . . . . . . . . . . . . . . . . . 19 ([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ 𝐴 / 𝑥𝑧𝐴 / 𝑥(V × 𝑈))
23 csbxp 5788 . . . . . . . . . . . . . . . . . . . . 21 𝐴 / 𝑥(V × 𝑈) = (𝐴 / 𝑥V × 𝐴 / 𝑥𝑈)
2410xpeq1d 5718 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝑉 → (𝐴 / 𝑥V × 𝐴 / 𝑥𝑈) = (V × 𝐴 / 𝑥𝑈))
2523, 24eqtrid 2787 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑉𝐴 / 𝑥(V × 𝑈) = (V × 𝐴 / 𝑥𝑈))
2615, 25eleq12d 2833 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥(V × 𝑈) ↔ 𝑧 ∈ (V × 𝐴 / 𝑥𝑈)))
2722, 26bitrid 283 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ 𝑧 ∈ (V × 𝐴 / 𝑥𝑈)))
28 csbconstg 3927 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥 𝑛, (1st𝑧)⟩ = ⟨ 𝑛, (1st𝑧)⟩)
29 csbconstg 3927 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥𝑛, 𝑧⟩ = ⟨𝑛, 𝑧⟩)
3027, 28, 29ifbieq12d 4559 . . . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥if(𝑧 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩) = if(𝑧 ∈ (V × 𝐴 / 𝑥𝑈), ⟨ 𝑛, (1st𝑧)⟩, ⟨𝑛, 𝑧⟩))
3219, 20, 31ifbieq12d 4559 . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . 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 7508 . . . . . . . . . . . . 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 2775 . . . . . . . . . . . 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 4896 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 / 𝑥𝑁, 𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝐴 / 𝑥𝑦⟩)
37 csbconstg 3927 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
3837opeq2d 4885 . . . . . . . . . . . . 13 (𝐴𝑉 → ⟨𝐴 / 𝑥𝑁, 𝐴 / 𝑥𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝑦⟩)
3936, 38eqtrd 2775 . . . . . . . . . . . 12 (𝐴𝑉𝐴 / 𝑥𝑁, 𝑦⟩ = ⟨𝐴 / 𝑥𝑁, 𝑦⟩)
40 rdgeq12 8452 . . . . . . . . . . . 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 2775 . . . . . . . . . 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 6909 . . . . . . . . 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 2787 . . . . . . . 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 2746 . . . . . . 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 279 . . . . . 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, 47bitrid 283 . . . 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 4446 . . 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 37367 . . 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 2800 . 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 2787 1 (𝐴𝑉𝐴 / 𝑥(𝑈↑↑𝑁) = (𝐴 / 𝑥𝑈↑↑𝐴 / 𝑥𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  Vcvv 3478  [wsbc 3791  csb 3908  c0 4339  ifcif 4531  cop 4637   cuni 4912   × cxp 5687  cfv 6563  cmpo 7433  ωcom 7887  1st c1st 8011  reccrdg 8448  1oc1o 8498  ↑↑cfinxp 37366
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-finxp 37367
This theorem is referenced by: (None)
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