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Mirrors > Home > MPE Home > Th. List > csbdm | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
csbdm | ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbab 4388 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} | |
2 | sbcex2 3833 | . . . . 5 ⊢ ([𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]〈𝑦, 𝑤〉 ∈ 𝐵) | |
3 | sbcel2 4366 | . . . . . 6 ⊢ ([𝐴 / 𝑥]〈𝑦, 𝑤〉 ∈ 𝐵 ↔ 〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵) | |
4 | 3 | exbii 1844 | . . . . 5 ⊢ (∃𝑤[𝐴 / 𝑥]〈𝑦, 𝑤〉 ∈ 𝐵 ↔ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵) |
5 | 2, 4 | bitri 277 | . . . 4 ⊢ ([𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵 ↔ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵) |
6 | 5 | abbii 2886 | . . 3 ⊢ {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} |
7 | 1, 6 | eqtri 2844 | . 2 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} |
8 | dfdm3 5752 | . . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} | |
9 | 8 | csbeq2i 3890 | . 2 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} |
10 | dfdm3 5752 | . 2 ⊢ dom ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} | |
11 | 7, 9, 10 | 3eqtr4i 2854 | 1 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 [wsbc 3771 ⦋csb 3882 〈cop 4566 dom cdm 5549 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-nul 4291 df-br 5059 df-dm 5559 |
This theorem is referenced by: sbcfng 6505 |
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