Theorem bj-cnvcnv 38064

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.) (Proof shortened by BJ, 16-Oct-2026.) Prefer use of bj-cnvcnv2 38063. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-cnvcnv	⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))

Proof of Theorem bj-cnvcnv

Step	Hyp	Ref	Expression
1		bj-cnvcnv2 38063	. 2 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2		df-res 5662	. 2 ⊢ (𝐴 ↾ V) = (𝐴 ∩ (V × V))
3	1, 2	eqtri 2783	1 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3450   ∩ cin 4088   × cxp 5648  ^◡ccnv 5649   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-res 5662

This theorem is used by: (None)