![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3066 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵}) | |
2 | sbcexg 2967 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]〈𝑤, 𝑦〉 ∈ 𝐵)) | |
3 | sbcel2g 3028 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]〈𝑤, 𝑦〉 ∈ 𝐵 ↔ 〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵)) | |
4 | 3 | exbidv 1798 | . . . . 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 2258 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵}) |
7 | 1, 6 | eqtrd 2173 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵}) |
8 | dfrn3 4736 | . . 3 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} | |
9 | 8 | csbeq2i 3034 | . 2 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} |
10 | dfrn3 4736 | . 2 ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} | |
11 | 7, 9, 10 | 3eqtr4g 2198 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∃wex 1469 ∈ wcel 1481 {cab 2126 [wsbc 2913 ⦋csb 3007 〈cop 3535 ran crn 4548 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-cnv 4555 df-dm 4557 df-rn 4558 |
This theorem is referenced by: sbcfg 5279 |
Copyright terms: Public domain | W3C validator |