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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 4925 | . . . . 5 ⊢ Rel ◡◡𝐴 | |
2 | df-rel 4554 | . . . . 5 ⊢ (Rel ◡◡𝐴 ↔ ◡◡𝐴 ⊆ (V × V)) | |
3 | 1, 2 | mpbi 144 | . . . 4 ⊢ ◡◡𝐴 ⊆ (V × V) |
4 | relxp 4656 | . . . . 5 ⊢ Rel (V × V) | |
5 | dfrel2 4997 | . . . . 5 ⊢ (Rel (V × V) ↔ ◡◡(V × V) = (V × V)) | |
6 | 4, 5 | mpbi 144 | . . . 4 ⊢ ◡◡(V × V) = (V × V) |
7 | 3, 6 | sseqtrri 3137 | . . 3 ⊢ ◡◡𝐴 ⊆ ◡◡(V × V) |
8 | dfss 3090 | . . 3 ⊢ (◡◡𝐴 ⊆ ◡◡(V × V) ↔ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V))) | |
9 | 7, 8 | mpbi 144 | . 2 ⊢ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V)) |
10 | cnvin 4954 | . 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (◡◡𝐴 ∩ ◡◡(V × V)) | |
11 | cnvin 4954 | . . . 4 ⊢ ◡(𝐴 ∩ (V × V)) = (◡𝐴 ∩ ◡(V × V)) | |
12 | 11 | cnveqi 4722 | . . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = ◡(◡𝐴 ∩ ◡(V × V)) |
13 | inss2 3302 | . . . . 5 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V) | |
14 | df-rel 4554 | . . . . 5 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V)) | |
15 | 13, 14 | mpbir 145 | . . . 4 ⊢ Rel (𝐴 ∩ (V × V)) |
16 | dfrel2 4997 | . . . 4 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))) | |
17 | 15, 16 | mpbi 144 | . . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)) |
18 | 12, 17 | eqtr3i 2163 | . 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (𝐴 ∩ (V × V)) |
19 | 9, 10, 18 | 3eqtr2i 2167 | 1 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 Vcvv 2689 ∩ cin 3075 ⊆ wss 3076 × cxp 4545 ◡ccnv 4546 Rel wrel 4552 |
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-ral 2422 df-rex 2423 df-v 2691 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-xp 4553 df-rel 4554 df-cnv 4555 |
This theorem is referenced by: cnvcnv2 5000 cnvcnvss 5001 structcnvcnv 12014 strslfv2d 12040 |
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