| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 4440 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} | |
| 2 | sbcex2 3850 | . . . . 5 ⊢ ([𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]〈𝑦, 𝑤〉 ∈ 𝐵) | |
| 3 | sbcel2 4418 | . . . . . 6 ⊢ ([𝐴 / 𝑥]〈𝑦, 𝑤〉 ∈ 𝐵 ↔ 〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵) | |
| 4 | 3 | exbii 1848 | . . . . 5 ⊢ (∃𝑤[𝐴 / 𝑥]〈𝑦, 𝑤〉 ∈ 𝐵 ↔ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵) |
| 5 | 2, 4 | bitri 275 | . . . 4 ⊢ ([𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵 ↔ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵) |
| 6 | 5 | abbii 2809 | . . 3 ⊢ {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} |
| 7 | 1, 6 | eqtri 2765 | . 2 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} |
| 8 | dfdm3 5898 | . . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} | |
| 9 | 8 | csbeq2i 3907 | . 2 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ 𝐵} |
| 10 | dfdm3 5898 | . 2 ⊢ dom ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤〈𝑦, 𝑤〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} | |
| 11 | 7, 9, 10 | 3eqtr4i 2775 | 1 ⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 [wsbc 3788 ⦋csb 3899 〈cop 4632 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-nul 4334 df-br 5144 df-dm 5695 |
| This theorem is referenced by: sbcfng 6733 |
| Copyright terms: Public domain | W3C validator |