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| Mirrors > Home > MPE Home > Th. List > cnvcnv | Structured version Visualization version GIF version | ||
| Description: The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.) (Proof shortened by BJ, 26-Nov-2021.) |
| Ref | Expression |
|---|---|
| cnvcnv | ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnvin 6164 | . . 3 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (◡◡𝐴 ∩ ◡◡(V × V)) | |
| 2 | cnvin 6164 | . . . 4 ⊢ ◡(𝐴 ∩ (V × V)) = (◡𝐴 ∩ ◡(V × V)) | |
| 3 | 2 | cnveqi 5885 | . . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = ◡(◡𝐴 ∩ ◡(V × V)) |
| 4 | relcnv 6122 | . . . . . 6 ⊢ Rel ◡◡𝐴 | |
| 5 | df-rel 5692 | . . . . . 6 ⊢ (Rel ◡◡𝐴 ↔ ◡◡𝐴 ⊆ (V × V)) | |
| 6 | 4, 5 | mpbi 230 | . . . . 5 ⊢ ◡◡𝐴 ⊆ (V × V) |
| 7 | relxp 5703 | . . . . . 6 ⊢ Rel (V × V) | |
| 8 | dfrel2 6209 | . . . . . 6 ⊢ (Rel (V × V) ↔ ◡◡(V × V) = (V × V)) | |
| 9 | 7, 8 | mpbi 230 | . . . . 5 ⊢ ◡◡(V × V) = (V × V) |
| 10 | 6, 9 | sseqtrri 4033 | . . . 4 ⊢ ◡◡𝐴 ⊆ ◡◡(V × V) |
| 11 | dfss 3970 | . . . 4 ⊢ (◡◡𝐴 ⊆ ◡◡(V × V) ↔ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V))) | |
| 12 | 10, 11 | mpbi 230 | . . 3 ⊢ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V)) |
| 13 | 1, 3, 12 | 3eqtr4ri 2776 | . 2 ⊢ ◡◡𝐴 = ◡◡(𝐴 ∩ (V × V)) |
| 14 | relinxp 5824 | . . 3 ⊢ Rel (𝐴 ∩ (V × V)) | |
| 15 | dfrel2 6209 | . . 3 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))) | |
| 16 | 14, 15 | mpbi 230 | . 2 ⊢ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)) |
| 17 | 13, 16 | eqtri 2765 | 1 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 × cxp 5683 ◡ccnv 5684 Rel wrel 5690 |
| 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: cnvcnv2 6213 cnvcnvss 6214 cnvrescnv 6215 structcnvcnv 17190 strfv2d 17238 elcnvcnvintab 43595 relintab 43596 nonrel 43597 elcnvcnvlem 43612 cnvcnvintabd 43613 tposresg 48778 |
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