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Mirrors > Home > ILE Home > Th. List > csbcnvg | GIF version |
Description: Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.) |
Ref | Expression |
---|---|
csbcnvg | ⊢ (A ∈ 𝑉 → ◡⦋A / x⦌𝐹 = ⦋A / x⦌◡𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcbrg 3804 | . . . . 5 ⊢ (A ∈ 𝑉 → ([A / x]z𝐹y ↔ ⦋A / x⦌z⦋A / x⦌𝐹⦋A / x⦌y)) | |
2 | csbconstg 2858 | . . . . . 6 ⊢ (A ∈ 𝑉 → ⦋A / x⦌z = z) | |
3 | csbconstg 2858 | . . . . . 6 ⊢ (A ∈ 𝑉 → ⦋A / x⦌y = y) | |
4 | 2, 3 | breq12d 3768 | . . . . 5 ⊢ (A ∈ 𝑉 → (⦋A / x⦌z⦋A / x⦌𝐹⦋A / x⦌y ↔ z⦋A / x⦌𝐹y)) |
5 | 1, 4 | bitrd 177 | . . . 4 ⊢ (A ∈ 𝑉 → ([A / x]z𝐹y ↔ z⦋A / x⦌𝐹y)) |
6 | 5 | opabbidv 3814 | . . 3 ⊢ (A ∈ 𝑉 → {〈y, z〉 ∣ [A / x]z𝐹y} = {〈y, z〉 ∣ z⦋A / x⦌𝐹y}) |
7 | csbopabg 3826 | . . 3 ⊢ (A ∈ 𝑉 → ⦋A / x⦌{〈y, z〉 ∣ z𝐹y} = {〈y, z〉 ∣ [A / x]z𝐹y}) | |
8 | df-cnv 4296 | . . . 4 ⊢ ◡⦋A / x⦌𝐹 = {〈y, z〉 ∣ z⦋A / x⦌𝐹y} | |
9 | 8 | a1i 9 | . . 3 ⊢ (A ∈ 𝑉 → ◡⦋A / x⦌𝐹 = {〈y, z〉 ∣ z⦋A / x⦌𝐹y}) |
10 | 6, 7, 9 | 3eqtr4rd 2080 | . 2 ⊢ (A ∈ 𝑉 → ◡⦋A / x⦌𝐹 = ⦋A / x⦌{〈y, z〉 ∣ z𝐹y}) |
11 | df-cnv 4296 | . . 3 ⊢ ◡𝐹 = {〈y, z〉 ∣ z𝐹y} | |
12 | 11 | csbeq2i 2870 | . 2 ⊢ ⦋A / x⦌◡𝐹 = ⦋A / x⦌{〈y, z〉 ∣ z𝐹y} |
13 | 10, 12 | syl6eqr 2087 | 1 ⊢ (A ∈ 𝑉 → ◡⦋A / x⦌𝐹 = ⦋A / x⦌◡𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 [wsbc 2758 ⦋csb 2846 class class class wbr 3755 {copab 3808 ◡ccnv 4287 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-cnv 4296 |
This theorem is referenced by: (None) |
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