Theorem csbdmg 4949

Description: Distribute proper substitution through the domain of a class. (Contributed by Jim Kingdon, 8-Dec-2018.)

Assertion

Ref	Expression
csbdmg	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵)

Proof of Theorem csbdmg

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

Step	Hyp	Ref	Expression
1		csbabg 3199	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵})
2		sbcex2 3095	. . . . 5 ⊢ ([𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]⟨𝑦, 𝑤⟩ ∈ 𝐵)
3		sbcel2g 3158	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵))
4	3	exbidv 1874	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∃𝑤[𝐴 / 𝑥]⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵))
5	2, 4	bitrid 192	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵))
6	5	abbidv 2352	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵})
7	1, 6	eqtrd 2265	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵})
8		dfdm3 4941	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵}
9	8	csbeq2i 3164	. 2 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵}
10		dfdm3 4941	. 2 ⊢ dom ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
11	7, 9, 10	3eqtr4g 2290	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   = wceq 1398  ∃wex 1541   ∈ wcel 2203  {cab 2218  [wsbc 3041  ⦋csb 3137  ⟨cop 3691  dom cdm 4748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-sbc 3042  df-csb 3138  df-br 4109  df-dm 4758

This theorem is referenced by:  sbcfng  5505