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Mirrors > Home > ILE Home > Th. List > cnvsym | GIF version |
Description: Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvsym | ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀x∀y(x𝑅y → y𝑅x)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alcom 1364 | . 2 ⊢ (∀y∀x(〈y, x〉 ∈ ◡𝑅 → 〈y, x〉 ∈ 𝑅) ↔ ∀x∀y(〈y, x〉 ∈ ◡𝑅 → 〈y, x〉 ∈ 𝑅)) | |
2 | relcnv 4646 | . . 3 ⊢ Rel ◡𝑅 | |
3 | ssrel 4371 | . . 3 ⊢ (Rel ◡𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ∀y∀x(〈y, x〉 ∈ ◡𝑅 → 〈y, x〉 ∈ 𝑅))) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀y∀x(〈y, x〉 ∈ ◡𝑅 → 〈y, x〉 ∈ 𝑅)) |
5 | vex 2554 | . . . . . 6 ⊢ y ∈ V | |
6 | vex 2554 | . . . . . 6 ⊢ x ∈ V | |
7 | 5, 6 | brcnv 4461 | . . . . 5 ⊢ (y◡𝑅x ↔ x𝑅y) |
8 | df-br 3756 | . . . . 5 ⊢ (y◡𝑅x ↔ 〈y, x〉 ∈ ◡𝑅) | |
9 | 7, 8 | bitr3i 175 | . . . 4 ⊢ (x𝑅y ↔ 〈y, x〉 ∈ ◡𝑅) |
10 | df-br 3756 | . . . 4 ⊢ (y𝑅x ↔ 〈y, x〉 ∈ 𝑅) | |
11 | 9, 10 | imbi12i 228 | . . 3 ⊢ ((x𝑅y → y𝑅x) ↔ (〈y, x〉 ∈ ◡𝑅 → 〈y, x〉 ∈ 𝑅)) |
12 | 11 | 2albii 1357 | . 2 ⊢ (∀x∀y(x𝑅y → y𝑅x) ↔ ∀x∀y(〈y, x〉 ∈ ◡𝑅 → 〈y, x〉 ∈ 𝑅)) |
13 | 1, 4, 12 | 3bitr4i 201 | 1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀x∀y(x𝑅y → y𝑅x)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1240 ∈ wcel 1390 ⊆ wss 2911 〈cop 3370 class class class wbr 3755 ◡ccnv 4287 Rel wrel 4293 |
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-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 |
This theorem is referenced by: dfer2 6043 |
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