Theorem imacnvcnv 6153

Description: The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
imacnvcnv	⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵)

Proof of Theorem imacnvcnv

Step	Hyp	Ref	Expression
1		rescnvcnv 6151	. . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
2	1	rneqi 5876	. 2 ⊢ ran (◡◡𝐴 ↾ 𝐵) = ran (𝐴 ↾ 𝐵)
3		df-ima 5627	. 2 ⊢ (◡◡𝐴 “ 𝐵) = ran (◡◡𝐴 ↾ 𝐵)
4		df-ima 5627	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
5	2, 3, 4	3eqtr4i 2764	1 ⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵)

Syntax hints:   = wceq 1541  ^◡ccnv 5613  ran crn 5615   ↾ cres 5616   “ cima 5617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627

This theorem is referenced by:  curry1  8034  curry2  8037  fnwelem  8061  fpwwe2lem5  10526  fpwwe2lem8  10529  eqglact  19091  hmeoima  23680  hmeocld  23682  hmeocls  23683  hmeontr  23684  reghmph  23708  qtopf1  23731  tgpconncompeqg  24027  imasf1obl  24403  mbfimaopnlem  25583  hmeoclda  36377