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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 5002 | . . . . 5 ⊢ Rel ◡◡𝐴 | |
2 | df-rel 4630 | . . . . 5 ⊢ (Rel ◡◡𝐴 ↔ ◡◡𝐴 ⊆ (V × V)) | |
3 | 1, 2 | mpbi 145 | . . . 4 ⊢ ◡◡𝐴 ⊆ (V × V) |
4 | relxp 4732 | . . . . 5 ⊢ Rel (V × V) | |
5 | dfrel2 5075 | . . . . 5 ⊢ (Rel (V × V) ↔ ◡◡(V × V) = (V × V)) | |
6 | 4, 5 | mpbi 145 | . . . 4 ⊢ ◡◡(V × V) = (V × V) |
7 | 3, 6 | sseqtrri 3190 | . . 3 ⊢ ◡◡𝐴 ⊆ ◡◡(V × V) |
8 | dfss 3143 | . . 3 ⊢ (◡◡𝐴 ⊆ ◡◡(V × V) ↔ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V))) | |
9 | 7, 8 | mpbi 145 | . 2 ⊢ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V)) |
10 | cnvin 5032 | . 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (◡◡𝐴 ∩ ◡◡(V × V)) | |
11 | cnvin 5032 | . . . 4 ⊢ ◡(𝐴 ∩ (V × V)) = (◡𝐴 ∩ ◡(V × V)) | |
12 | 11 | cnveqi 4798 | . . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = ◡(◡𝐴 ∩ ◡(V × V)) |
13 | inss2 3356 | . . . . 5 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V) | |
14 | df-rel 4630 | . . . . 5 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V)) | |
15 | 13, 14 | mpbir 146 | . . . 4 ⊢ Rel (𝐴 ∩ (V × V)) |
16 | dfrel2 5075 | . . . 4 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))) | |
17 | 15, 16 | mpbi 145 | . . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)) |
18 | 12, 17 | eqtr3i 2200 | . 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (𝐴 ∩ (V × V)) |
19 | 9, 10, 18 | 3eqtr2i 2204 | 1 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 Vcvv 2737 ∩ cin 3128 ⊆ wss 3129 × cxp 4621 ◡ccnv 4622 Rel wrel 4628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-xp 4629 df-rel 4630 df-cnv 4631 |
This theorem is referenced by: cnvcnv2 5078 cnvcnvss 5079 structcnvcnv 12458 strslfv2d 12484 |
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