Theorem csbdm 5806

Description: Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.) (Revised by NM, 24-Aug-2018.)

Assertion

Ref	Expression
csbdm	⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵

Proof of Theorem csbdm

Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbab 4371	. . 3 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵}
2		sbcex2 3781	. . . . 5 ⊢ ([𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]⟨𝑦, 𝑤⟩ ∈ 𝐵)
3		sbcel2 4349	. . . . . 6 ⊢ ([𝐴 / 𝑥]⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)
4	3	exbii 1850	. . . . 5 ⊢ (∃𝑤[𝐴 / 𝑥]⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)
5	2, 4	bitri 274	. . . 4 ⊢ ([𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)
6	5	abbii 2808	. . 3 ⊢ {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
7	1, 6	eqtri 2766	. 2 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
8		dfdm3 5796	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵}
9	8	csbeq2i 3840	. 2 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵}
10		dfdm3 5796	. 2 ⊢ dom ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
11	7, 9, 10	3eqtr4i 2776	1 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵

Syntax hints:   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  [wsbc 3716  ⦋csb 3832  ⟨cop 4567  dom cdm 5589

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-br 5075  df-dm 5599

This theorem is referenced by:  sbcfng  6597