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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 6167 | . . 3 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (◡◡𝐴 ∩ ◡◡(V × V)) | |
2 | cnvin 6167 | . . . 4 ⊢ ◡(𝐴 ∩ (V × V)) = (◡𝐴 ∩ ◡(V × V)) | |
3 | 2 | cnveqi 5888 | . . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = ◡(◡𝐴 ∩ ◡(V × V)) |
4 | relcnv 6125 | . . . . . 6 ⊢ Rel ◡◡𝐴 | |
5 | df-rel 5696 | . . . . . 6 ⊢ (Rel ◡◡𝐴 ↔ ◡◡𝐴 ⊆ (V × V)) | |
6 | 4, 5 | mpbi 230 | . . . . 5 ⊢ ◡◡𝐴 ⊆ (V × V) |
7 | relxp 5707 | . . . . . 6 ⊢ Rel (V × V) | |
8 | dfrel2 6211 | . . . . . 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 3982 | . . . 4 ⊢ (◡◡𝐴 ⊆ ◡◡(V × V) ↔ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V))) | |
12 | 10, 11 | mpbi 230 | . . 3 ⊢ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V)) |
13 | 1, 3, 12 | 3eqtr4ri 2774 | . 2 ⊢ ◡◡𝐴 = ◡◡(𝐴 ∩ (V × V)) |
14 | relinxp 5827 | . . 3 ⊢ Rel (𝐴 ∩ (V × V)) | |
15 | dfrel2 6211 | . . 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 2763 | 1 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 × cxp 5687 ◡ccnv 5688 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 |
This theorem is referenced by: cnvcnv2 6215 cnvcnvss 6216 cnvrescnv 6217 structcnvcnv 17187 strfv2d 17236 elcnvcnvintab 43572 relintab 43573 nonrel 43574 elcnvcnvlem 43589 cnvcnvintabd 43590 |
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