Theorem imacnvcnv 6228

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 6226	. . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
2	1	rneqi 5951	. 2 ⊢ ran (◡◡𝐴 ↾ 𝐵) = ran (𝐴 ↾ 𝐵)
3		df-ima 5702	. 2 ⊢ (◡◡𝐴 “ 𝐵) = ran (◡◡𝐴 ↾ 𝐵)
4		df-ima 5702	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
5	2, 3, 4	3eqtr4i 2773	1 ⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵)

Syntax hints:   = wceq 1537  ^◡ccnv 5688  ran crn 5690   ↾ cres 5691   “ cima 5692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702

This theorem is referenced by:  curry1  8128  curry2  8131  fnwelem  8155  fpwwe2lem5  10673  fpwwe2lem8  10676  eqglact  19210  hmeoima  23789  hmeocld  23791  hmeocls  23792  hmeontr  23793  reghmph  23817  qtopf1  23840  tgpconncompeqg  24136  imasf1obl  24517  mbfimaopnlem  25704  hmeoclda  36316