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Mirrors > Home > ILE Home > Th. List > csbdmg | GIF version |
Description: Distribute proper substitution through the domain of a class. (Contributed by Jim Kingdon, 8-Dec-2018.) |
Ref | Expression |
---|---|
csbdmg | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbabg 2907 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵}) | |
2 | sbcex2 2812 | . . . . 5 ⊢ ([𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]〈𝑦, 𝑤〉 ∈ 𝐵) | |
3 | sbcel2g 2871 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]〈𝑦, 𝑤〉 ∈ 𝐵 ↔ 〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵)) | |
4 | 3 | exbidv 1706 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∃𝑤[𝐴 / 𝑥]〈𝑦, 𝑤〉 ∈ 𝐵 ↔ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵)) |
5 | 2, 4 | syl5bb 181 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵 ↔ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵)) |
6 | 5 | abbidv 2155 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵}) |
7 | 1, 6 | eqtrd 2072 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵}) |
8 | dfdm3 4522 | . . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} | |
9 | 8 | csbeq2i 2876 | . 2 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} |
10 | dfdm3 4522 | . 2 ⊢ dom ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} | |
11 | 7, 9, 10 | 3eqtr4g 2097 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∃wex 1381 ∈ wcel 1393 {cab 2026 [wsbc 2764 ⦋csb 2852 〈cop 3378 dom cdm 4345 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-sbc 2765 df-csb 2853 df-br 3765 df-dm 4355 |
This theorem is referenced by: sbcfng 5044 |
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