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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 3061 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵}) | |
2 | sbcexg 2963 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]〈𝑤, 𝑦〉 ∈ 𝐵)) | |
3 | sbcel2g 3023 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]〈𝑤, 𝑦〉 ∈ 𝐵 ↔ 〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵)) | |
4 | 3 | exbidv 1797 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∃𝑤[𝐴 / 𝑥]〈𝑤, 𝑦〉 ∈ 𝐵 ↔ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵)) |
5 | 2, 4 | bitrd 187 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵 ↔ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵)) |
6 | 5 | abbidv 2257 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵}) |
7 | 1, 6 | eqtrd 2172 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵}) |
8 | dfrn3 4728 | . . 3 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} | |
9 | 8 | csbeq2i 3029 | . 2 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} |
10 | dfrn3 4728 | . 2 ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} | |
11 | 7, 9, 10 | 3eqtr4g 2197 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∃wex 1468 ∈ wcel 1480 {cab 2125 [wsbc 2909 ⦋csb 3003 〈cop 3530 ran crn 4540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-cnv 4547 df-dm 4549 df-rn 4550 |
This theorem is referenced by: sbcfg 5271 |
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