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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 5575 | . . 3 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (◡◡𝐴 ∩ ◡◡(V × V)) | |
2 | cnvin 5575 | . . . 4 ⊢ ◡(𝐴 ∩ (V × V)) = (◡𝐴 ∩ ◡(V × V)) | |
3 | 2 | cnveqi 5329 | . . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = ◡(◡𝐴 ∩ ◡(V × V)) |
4 | relcnv 5538 | . . . . . 6 ⊢ Rel ◡◡𝐴 | |
5 | df-rel 5150 | . . . . . 6 ⊢ (Rel ◡◡𝐴 ↔ ◡◡𝐴 ⊆ (V × V)) | |
6 | 4, 5 | mpbi 220 | . . . . 5 ⊢ ◡◡𝐴 ⊆ (V × V) |
7 | relxp 5160 | . . . . . 6 ⊢ Rel (V × V) | |
8 | dfrel2 5618 | . . . . . 6 ⊢ (Rel (V × V) ↔ ◡◡(V × V) = (V × V)) | |
9 | 7, 8 | mpbi 220 | . . . . 5 ⊢ ◡◡(V × V) = (V × V) |
10 | 6, 9 | sseqtr4i 3671 | . . . 4 ⊢ ◡◡𝐴 ⊆ ◡◡(V × V) |
11 | dfss 3622 | . . . 4 ⊢ (◡◡𝐴 ⊆ ◡◡(V × V) ↔ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V))) | |
12 | 10, 11 | mpbi 220 | . . 3 ⊢ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V)) |
13 | 1, 3, 12 | 3eqtr4ri 2684 | . 2 ⊢ ◡◡𝐴 = ◡◡(𝐴 ∩ (V × V)) |
14 | inss2 3867 | . . . 4 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V) | |
15 | df-rel 5150 | . . . 4 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V)) | |
16 | 14, 15 | mpbir 221 | . . 3 ⊢ Rel (𝐴 ∩ (V × V)) |
17 | dfrel2 5618 | . . 3 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))) | |
18 | 16, 17 | mpbi 220 | . 2 ⊢ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)) |
19 | 13, 18 | eqtri 2673 | 1 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 Vcvv 3231 ∩ cin 3606 ⊆ wss 3607 × cxp 5141 ◡ccnv 5142 Rel wrel 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-xp 5149 df-rel 5150 df-cnv 5151 |
This theorem is referenced by: cnvcnv2 5623 cnvcnvss 5624 structcnvcnv 15918 strfv2d 15952 elcnvcnvintab 38205 relintab 38206 nonrel 38207 elcnvcnvlem 38222 cnvcnvintabd 38223 |
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