MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvcnv Structured version   Visualization version   GIF version

Theorem cnvcnv 5727
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 5681 . . 3 (𝐴(V × V)) = (𝐴(V × V))
2 cnvin 5681 . . . 4 (𝐴 ∩ (V × V)) = (𝐴(V × V))
32cnveqi 5435 . . 3 (𝐴 ∩ (V × V)) = (𝐴(V × V))
4 relcnv 5644 . . . . . 6 Rel 𝐴
5 df-rel 5256 . . . . . 6 (Rel 𝐴𝐴 ⊆ (V × V))
64, 5mpbi 220 . . . . 5 𝐴 ⊆ (V × V)
7 relxp 5266 . . . . . 6 Rel (V × V)
8 dfrel2 5724 . . . . . 6 (Rel (V × V) ↔ (V × V) = (V × V))
97, 8mpbi 220 . . . . 5 (V × V) = (V × V)
106, 9sseqtr4i 3785 . . . 4 𝐴(V × V)
11 dfss 3736 . . . 4 (𝐴(V × V) ↔ 𝐴 = (𝐴(V × V)))
1210, 11mpbi 220 . . 3 𝐴 = (𝐴(V × V))
131, 3, 123eqtr4ri 2803 . 2 𝐴 = (𝐴 ∩ (V × V))
14 inss2 3980 . . . 4 (𝐴 ∩ (V × V)) ⊆ (V × V)
15 df-rel 5256 . . . 4 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
1614, 15mpbir 221 . . 3 Rel (𝐴 ∩ (V × V))
17 dfrel2 5724 . . 3 (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
1816, 17mpbi 220 . 2 (𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
1913, 18eqtri 2792 1 𝐴 = (𝐴 ∩ (V × V))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1630  Vcvv 3349  cin 3720  wss 3721   × cxp 5247  ccnv 5248  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-cnv 5257
This theorem is referenced by:  cnvcnv2  5729  cnvcnvss  5730  structcnvcnv  16077  strfv2d  16111  elcnvcnvintab  38407  relintab  38408  nonrel  38409  elcnvcnvlem  38424  cnvcnvintabd  38425
  Copyright terms: Public domain W3C validator