Theorem imacnvcnv 5227

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 5225	. . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
2	1	rneqi 4985	. 2 ⊢ ran (◡◡𝐴 ↾ 𝐵) = ran (𝐴 ↾ 𝐵)
3		df-ima 4762	. 2 ⊢ (◡◡𝐴 “ 𝐵) = ran (◡◡𝐴 ↾ 𝐵)
4		df-ima 4762	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
5	2, 3, 4	3eqtr4i 2263	1 ⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵)

Syntax hints:   = wceq 1398  ^◡ccnv 4748  ran crn 4750   ↾ cres 4751   “ cima 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762

This theorem is referenced by:  casedm  7377  caseinl  7382  caseinr  7383  djudm  7396  eqglact  13942  hmeoima  15175  hmeocld  15177  hmeontr  15178