Theorem csbmpo123 35502

Description: Move class substitution in and out of maps-to notation for operations. (Contributed by ML, 25-Oct-2020.)

Assertion

Ref	Expression
csbmpo123	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐷) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌, 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍 ↦ ⦋𝐴 / 𝑥⦌𝐷))

Distinct variable groups:   𝑦,𝐴   𝑧,𝐴   𝑦,𝑉   𝑧,𝑉   𝑥,𝑦   𝑥,𝑧

Allowed substitution hints:   𝐴(𝑥)   𝐷(𝑥,𝑦,𝑧)   𝑉(𝑥)   𝑌(𝑥,𝑦,𝑧)   𝑍(𝑥,𝑦,𝑧)

Proof of Theorem csbmpo123

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csboprabg 35501	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ 𝑑 = 𝐷)} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ 𝑑 = 𝐷)})
2		sbcan 3768	. . . . 5 ⊢ ([𝐴 / 𝑥]((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ 𝑑 = 𝐷) ↔ ([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ [𝐴 / 𝑥]𝑑 = 𝐷))
3		sbcan 3768	. . . . . . 7 ⊢ ([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ∧ [𝐴 / 𝑥]𝑧 ∈ 𝑍))
4		sbcel12 4342	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌)
5		csbconstg 3851	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
6	5	eleq1d 2823	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌))
7	4, 6	syl5bb 283	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌))
8		sbcel12 4342	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝑍 ↔ ⦋𝐴 / 𝑥⦌𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍)
9		csbconstg 3851	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
10	9	eleq1d 2823	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍 ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍))
11	8, 10	syl5bb 283	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ∈ 𝑍 ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍))
12	7, 11	anbi12d 631	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑦 ∈ 𝑌 ∧ [𝐴 / 𝑥]𝑧 ∈ 𝑍) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍)))
13	3, 12	syl5bb 283	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍)))
14		sbceq2g 4350	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑑 = 𝐷 ↔ 𝑑 = ⦋𝐴 / 𝑥⦌𝐷))
15	13, 14	anbi12d 631	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ [𝐴 / 𝑥]𝑑 = 𝐷) ↔ ((𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍) ∧ 𝑑 = ⦋𝐴 / 𝑥⦌𝐷)))
16	2, 15	syl5bb 283	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ 𝑑 = 𝐷) ↔ ((𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍) ∧ 𝑑 = ⦋𝐴 / 𝑥⦌𝐷)))
17	16	oprabbidv 7341	. . 3 ⊢ (𝐴 ∈ 𝑉 → {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ 𝑑 = 𝐷)} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍) ∧ 𝑑 = ⦋𝐴 / 𝑥⦌𝐷)})
18	1, 17	eqtrd 2778	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ 𝑑 = 𝐷)} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍) ∧ 𝑑 = ⦋𝐴 / 𝑥⦌𝐷)})
19		df-mpo 7280	. . 3 ⊢ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐷) = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ 𝑑 = 𝐷)}
20	19	csbeq2i 3840	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐷) = ⦋𝐴 / 𝑥⦌{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ 𝑑 = 𝐷)}
21		df-mpo 7280	. 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌, 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍 ↦ ⦋𝐴 / 𝑥⦌𝐷) = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ ((𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍) ∧ 𝑑 = ⦋𝐴 / 𝑥⦌𝐷)}
22	18, 20, 21	3eqtr4g 2803	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐷) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌, 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍 ↦ ⦋𝐴 / 𝑥⦌𝐷))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  [wsbc 3716  ⦋csb 3832  {coprab 7276   ∈ cmpo 7277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-nul 4257  df-oprab 7279  df-mpo 7280

This theorem is referenced by:  csbfinxpg  35559