Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4917 | . . . . 5 ⊢ Rel ◡◡𝐴 | |
2 | df-rel 4546 | . . . . 5 ⊢ (Rel ◡◡𝐴 ↔ ◡◡𝐴 ⊆ (V × V)) | |
3 | 1, 2 | mpbi 144 | . . . 4 ⊢ ◡◡𝐴 ⊆ (V × V) |
4 | relxp 4648 | . . . . 5 ⊢ Rel (V × V) | |
5 | dfrel2 4989 | . . . . 5 ⊢ (Rel (V × V) ↔ ◡◡(V × V) = (V × V)) | |
6 | 4, 5 | mpbi 144 | . . . 4 ⊢ ◡◡(V × V) = (V × V) |
7 | 3, 6 | sseqtrri 3132 | . . 3 ⊢ ◡◡𝐴 ⊆ ◡◡(V × V) |
8 | dfss 3085 | . . 3 ⊢ (◡◡𝐴 ⊆ ◡◡(V × V) ↔ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V))) | |
9 | 7, 8 | mpbi 144 | . 2 ⊢ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V)) |
10 | cnvin 4946 | . 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (◡◡𝐴 ∩ ◡◡(V × V)) | |
11 | cnvin 4946 | . . . 4 ⊢ ◡(𝐴 ∩ (V × V)) = (◡𝐴 ∩ ◡(V × V)) | |
12 | 11 | cnveqi 4714 | . . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = ◡(◡𝐴 ∩ ◡(V × V)) |
13 | inss2 3297 | . . . . 5 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V) | |
14 | df-rel 4546 | . . . . 5 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V)) | |
15 | 13, 14 | mpbir 145 | . . . 4 ⊢ Rel (𝐴 ∩ (V × V)) |
16 | dfrel2 4989 | . . . 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 2162 | . 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (𝐴 ∩ (V × V)) |
19 | 9, 10, 18 | 3eqtr2i 2166 | 1 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 Vcvv 2686 ∩ cin 3070 ⊆ wss 3071 × cxp 4537 ◡ccnv 4538 Rel wrel 4544 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 |
This theorem is referenced by: cnvcnv2 4992 cnvcnvss 4993 structcnvcnv 11978 strslfv2d 12004 |
Copyright terms: Public domain | W3C validator |