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| 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 6212 | . 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) | |
| 2 | df-res 5697 | . 2 ⊢ (𝐴 ↾ V) = (𝐴 ∩ (V × V)) | |
| 3 | 1, 2 | eqtr4i 2768 | 1 ⊢ ◡◡𝐴 = (𝐴 ↾ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 Vcvv 3480 ∩ cin 3950 × cxp 5683 ◡ccnv 5684 ↾ cres 5687 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-res 5697 | 
| This theorem is referenced by: dfrel3 6218 rnresv 6221 rescnvcnv 6224 cocnvcnv1 6277 cocnvcnv2 6278 strfv2d 17238 resnonrel 43605 | 
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