|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > cnvcnvss | Structured version Visualization version GIF version | ||
| Description: The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.) | 
| Ref | Expression | 
|---|---|
| cnvcnvss | ⊢ ◡◡𝐴 ⊆ 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cnvcnv 6212 | . 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) | |
| 2 | inss1 4237 | . 2 ⊢ (𝐴 ∩ (V × V)) ⊆ 𝐴 | |
| 3 | 1, 2 | eqsstri 4030 | 1 ⊢ ◡◡𝐴 ⊆ 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 × cxp 5683 ◡ccnv 5684 | 
| 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 | 
| This theorem is referenced by: funcnvcnv 6633 foimacnv 6865 cnvct 9074 cnvfiALT 9379 structcnvcnv 17190 mvdco 19463 fcoinver 32617 fcnvgreu 32683 cnvssb 43599 relnonrel 43600 clcnvlem 43636 cnvtrrel 43683 relexpaddss 43731 tposres3 48781 | 
| Copyright terms: Public domain | W3C validator |