Theorem imacnvcnv 6206

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 6204	. . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
2	1	rneqi 5925	. 2 ⊢ ran (◡◡𝐴 ↾ 𝐵) = ran (𝐴 ↾ 𝐵)
3		df-ima 5672	. 2 ⊢ (◡◡𝐴 “ 𝐵) = ran (◡◡𝐴 ↾ 𝐵)
4		df-ima 5672	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
5	2, 3, 4	3eqtr4i 2795	1 ⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵)

This proof depends on syntax axioms:   = wceq 1570  ^◡ccnv 5658  ran crn 5660   ↾ cres 5661   “ cima 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672

This theorem is used by:  curry1  8105  curry2  8108  fnwelem  8133  fpwwe2lem5  10648  fpwwe2lem8  10651  eqglact  19310  hmeoima  23997  hmeocld  23999  hmeocls  24000  hmeontr  24001  reghmph  24025  qtopf1  24048  tgpconncompeqg  24344  imasf1obl  24720  mbfimaopnlem  25889  hmeoclda  36960