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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 2963 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵}) | |
2 | sbcex2 2867 | . . . . 5 ⊢ ([𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]〈𝑦, 𝑤〉 ∈ 𝐵) | |
3 | sbcel2g 2927 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]〈𝑦, 𝑤〉 ∈ 𝐵 ↔ 〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵)) | |
4 | 3 | exbidv 1746 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∃𝑤[𝐴 / 𝑥]〈𝑦, 𝑤〉 ∈ 𝐵 ↔ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵)) |
5 | 2, 4 | syl5bb 190 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵 ↔ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵)) |
6 | 5 | abbidv 2196 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵}) |
7 | 1, 6 | eqtrd 2113 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵}) |
8 | dfdm3 4540 | . . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} | |
9 | 8 | csbeq2i 2932 | . 2 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} |
10 | dfdm3 4540 | . 2 ⊢ dom ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} | |
11 | 7, 9, 10 | 3eqtr4g 2138 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∃wex 1421 ∈ wcel 1433 {cab 2067 [wsbc 2815 ⦋csb 2908 〈cop 3401 dom cdm 4363 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 df-csb 2909 df-br 3786 df-dm 4373 |
This theorem is referenced by: sbcfng 5064 |
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