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Theorem cnvcnv 5621
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.)
Assertion
Ref Expression
cnvcnv 𝐴 = (𝐴 ∩ (V × V))

Proof of Theorem cnvcnv
StepHypRef Expression
1 cnvin 5575 . . 3 (𝐴(V × V)) = (𝐴(V × V))
2 cnvin 5575 . . . 4 (𝐴 ∩ (V × V)) = (𝐴(V × V))
32cnveqi 5329 . . 3 (𝐴 ∩ (V × V)) = (𝐴(V × V))
4 relcnv 5538 . . . . . 6 Rel 𝐴
5 df-rel 5150 . . . . . 6 (Rel 𝐴𝐴 ⊆ (V × V))
64, 5mpbi 220 . . . . 5 𝐴 ⊆ (V × V)
7 relxp 5160 . . . . . 6 Rel (V × V)
8 dfrel2 5618 . . . . . 6 (Rel (V × V) ↔ (V × V) = (V × V))
97, 8mpbi 220 . . . . 5 (V × V) = (V × V)
106, 9sseqtr4i 3671 . . . 4 𝐴(V × V)
11 dfss 3622 . . . 4 (𝐴(V × V) ↔ 𝐴 = (𝐴(V × V)))
1210, 11mpbi 220 . . 3 𝐴 = (𝐴(V × V))
131, 3, 123eqtr4ri 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))
1614, 15mpbir 221 . . 3 Rel (𝐴 ∩ (V × V))
17 dfrel2 5618 . . 3 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
1816, 17mpbi 220 . 2 (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
1913, 18eqtri 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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