Theorem csbdm 5899

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 4438	. . 3 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵}
2		sbcex2 3839	. . . . 5 ⊢ ([𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]⟨𝑦, 𝑤⟩ ∈ 𝐵)
3		sbcel2 4416	. . . . . 6 ⊢ ([𝐴 / 𝑥]⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)
4	3	exbii 1842	. . . . 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 2795	. . 3 ⊢ {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
7	1, 6	eqtri 2753	. 2 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
8		dfdm3 5889	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵}
9	8	csbeq2i 3898	. 2 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵}
10		dfdm3 5889	. 2 ⊢ dom ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
11	7, 9, 10	3eqtr4i 2763	1 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵

Syntax hints:   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702  [wsbc 3774  ⦋csb 3890  ⟨cop 4635  dom cdm 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-nul 4324  df-br 5149  df-dm 5687

This theorem is referenced by:  sbcfng  6718