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Mirrors > Home > ILE Home > Th. List > cnvcnv2 | GIF version |
Description: The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.) |
Ref | Expression |
---|---|
cnvcnv2 | ⊢ ◡◡𝐴 = (𝐴 ↾ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnv 5077 | . 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) | |
2 | df-res 4635 | . 2 ⊢ (𝐴 ↾ V) = (𝐴 ∩ (V × V)) | |
3 | 1, 2 | eqtr4i 2201 | 1 ⊢ ◡◡𝐴 = (𝐴 ↾ V) |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 Vcvv 2737 ∩ cin 3128 × cxp 4621 ◡ccnv 4622 ↾ cres 4625 |
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 df-res 4635 |
This theorem is referenced by: dfrel3 5082 rnresv 5084 rescnvcnv 5087 cocnvcnv1 5135 cocnvcnv2 5136 strslfv2d 12484 |
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