Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5998 | . . 3 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (◡◡𝐴 ∩ ◡◡(V × V)) | |
2 | cnvin 5998 | . . . 4 ⊢ ◡(𝐴 ∩ (V × V)) = (◡𝐴 ∩ ◡(V × V)) | |
3 | 2 | cnveqi 5740 | . . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = ◡(◡𝐴 ∩ ◡(V × V)) |
4 | relcnv 5962 | . . . . . 6 ⊢ Rel ◡◡𝐴 | |
5 | df-rel 5557 | . . . . . 6 ⊢ (Rel ◡◡𝐴 ↔ ◡◡𝐴 ⊆ (V × V)) | |
6 | 4, 5 | mpbi 232 | . . . . 5 ⊢ ◡◡𝐴 ⊆ (V × V) |
7 | relxp 5568 | . . . . . 6 ⊢ Rel (V × V) | |
8 | dfrel2 6041 | . . . . . 6 ⊢ (Rel (V × V) ↔ ◡◡(V × V) = (V × V)) | |
9 | 7, 8 | mpbi 232 | . . . . 5 ⊢ ◡◡(V × V) = (V × V) |
10 | 6, 9 | sseqtrri 4004 | . . . 4 ⊢ ◡◡𝐴 ⊆ ◡◡(V × V) |
11 | dfss 3953 | . . . 4 ⊢ (◡◡𝐴 ⊆ ◡◡(V × V) ↔ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V))) | |
12 | 10, 11 | mpbi 232 | . . 3 ⊢ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V)) |
13 | 1, 3, 12 | 3eqtr4ri 2855 | . 2 ⊢ ◡◡𝐴 = ◡◡(𝐴 ∩ (V × V)) |
14 | relinxp 5682 | . . 3 ⊢ Rel (𝐴 ∩ (V × V)) | |
15 | dfrel2 6041 | . . 3 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))) | |
16 | 14, 15 | mpbi 232 | . 2 ⊢ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)) |
17 | 13, 16 | eqtri 2844 | 1 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3495 ∩ cin 3935 ⊆ wss 3936 × cxp 5548 ◡ccnv 5549 Rel wrel 5555 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-cnv 5558 |
This theorem is referenced by: cnvcnv2 6045 cnvcnvss 6046 cnvrescnv 6047 structcnvcnv 16491 strfv2d 16523 elcnvcnvintab 39935 relintab 39936 nonrel 39937 elcnvcnvlem 39952 cnvcnvintabd 39953 |
Copyright terms: Public domain | W3C validator |