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Mirrors > Home > ILE Home > Th. List > csbrng | GIF version |
Description: Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.) |
Ref | Expression |
---|---|
csbrng | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbabg 2963 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵}) | |
2 | sbcexg 2868 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]〈𝑤, 𝑦〉 ∈ 𝐵)) | |
3 | sbcel2g 2927 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]〈𝑤, 𝑦〉 ∈ 𝐵 ↔ 〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵)) | |
4 | 3 | exbidv 1746 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∃𝑤[𝐴 / 𝑥]〈𝑤, 𝑦〉 ∈ 𝐵 ↔ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵)) |
5 | 2, 4 | bitrd 186 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵 ↔ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵)) |
6 | 5 | abbidv 2196 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵}) |
7 | 1, 6 | eqtrd 2113 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵}) |
8 | dfrn3 4542 | . . 3 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} | |
9 | 8 | csbeq2i 2932 | . 2 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} |
10 | dfrn3 4542 | . 2 ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} | |
11 | 7, 9, 10 | 3eqtr4g 2138 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋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 ran crn 4364 |
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-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-cnv 4371 df-dm 4373 df-rn 4374 |
This theorem is referenced by: sbcfg 5065 |
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