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Mirrors > Home > ILE Home > Th. List > cnvcnv | GIF version |
Description: The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.) |
Ref | Expression |
---|---|
cnvcnv | ⊢ ◡◡A = (A ∩ (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4646 | . . . . 5 ⊢ Rel ◡◡A | |
2 | df-rel 4295 | . . . . 5 ⊢ (Rel ◡◡A ↔ ◡◡A ⊆ (V × V)) | |
3 | 1, 2 | mpbi 133 | . . . 4 ⊢ ◡◡A ⊆ (V × V) |
4 | relxp 4390 | . . . . 5 ⊢ Rel (V × V) | |
5 | dfrel2 4714 | . . . . 5 ⊢ (Rel (V × V) ↔ ◡◡(V × V) = (V × V)) | |
6 | 4, 5 | mpbi 133 | . . . 4 ⊢ ◡◡(V × V) = (V × V) |
7 | 3, 6 | sseqtr4i 2972 | . . 3 ⊢ ◡◡A ⊆ ◡◡(V × V) |
8 | dfss 2926 | . . 3 ⊢ (◡◡A ⊆ ◡◡(V × V) ↔ ◡◡A = (◡◡A ∩ ◡◡(V × V))) | |
9 | 7, 8 | mpbi 133 | . 2 ⊢ ◡◡A = (◡◡A ∩ ◡◡(V × V)) |
10 | cnvin 4674 | . 2 ⊢ ◡(◡A ∩ ◡(V × V)) = (◡◡A ∩ ◡◡(V × V)) | |
11 | cnvin 4674 | . . . 4 ⊢ ◡(A ∩ (V × V)) = (◡A ∩ ◡(V × V)) | |
12 | 11 | cnveqi 4453 | . . 3 ⊢ ◡◡(A ∩ (V × V)) = ◡(◡A ∩ ◡(V × V)) |
13 | inss2 3152 | . . . . 5 ⊢ (A ∩ (V × V)) ⊆ (V × V) | |
14 | df-rel 4295 | . . . . 5 ⊢ (Rel (A ∩ (V × V)) ↔ (A ∩ (V × V)) ⊆ (V × V)) | |
15 | 13, 14 | mpbir 134 | . . . 4 ⊢ Rel (A ∩ (V × V)) |
16 | dfrel2 4714 | . . . 4 ⊢ (Rel (A ∩ (V × V)) ↔ ◡◡(A ∩ (V × V)) = (A ∩ (V × V))) | |
17 | 15, 16 | mpbi 133 | . . 3 ⊢ ◡◡(A ∩ (V × V)) = (A ∩ (V × V)) |
18 | 12, 17 | eqtr3i 2059 | . 2 ⊢ ◡(◡A ∩ ◡(V × V)) = (A ∩ (V × V)) |
19 | 9, 10, 18 | 3eqtr2i 2063 | 1 ⊢ ◡◡A = (A ∩ (V × V)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 Vcvv 2551 ∩ cin 2910 ⊆ wss 2911 × cxp 4286 ◡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: cnvcnv2 4717 cnvcnvss 4718 |
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