Theorem csbdm 5911

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 4446	. . 3 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵}
2		sbcex2 3856	. . . . 5 ⊢ ([𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]⟨𝑦, 𝑤⟩ ∈ 𝐵)
3		sbcel2 4424	. . . . . 6 ⊢ ([𝐴 / 𝑥]⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)
4	3	exbii 1845	. . . . 5 ⊢ (∃𝑤[𝐴 / 𝑥]⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)
5	2, 4	bitri 275	. . . 4 ⊢ ([𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)
6	5	abbii 2807	. . 3 ⊢ {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
7	1, 6	eqtri 2763	. 2 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
8		dfdm3 5901	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵}
9	8	csbeq2i 3916	. 2 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵}
10		dfdm3 5901	. 2 ⊢ dom ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
11	7, 9, 10	3eqtr4i 2773	1 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵

Syntax hints:   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712  [wsbc 3791  ⦋csb 3908  ⟨cop 4637  dom cdm 5689

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-nul 4340  df-br 5149  df-dm 5699

This theorem is referenced by:  sbcfng  6734