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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 4437 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} | |
2 | sbcex2 3842 | . . . . 5 ⊢ ([𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]⟨𝑦, 𝑤⟩ ∈ 𝐵) | |
3 | sbcel2 4415 | . . . . . 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 2802 | . . 3 ⊢ {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} |
7 | 1, 6 | eqtri 2760 | . 2 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} |
8 | dfdm3 5887 | . . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} | |
9 | 8 | csbeq2i 3901 | . 2 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ 𝐵} |
10 | dfdm3 5887 | . 2 ⊢ dom ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑦, 𝑤⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} | |
11 | 7, 9, 10 | 3eqtr4i 2770 | 1 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2709 [wsbc 3777 ⦋csb 3893 ⟨cop 4634 dom cdm 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-nul 4323 df-br 5149 df-dm 5686 |
This theorem is referenced by: sbcfng 6714 |
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