Theorem imadmres 6237

Description: The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
imadmres	⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵)

Proof of Theorem imadmres

Step	Hyp	Ref	Expression
1		resdmres 6235	. . 3 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵)
2	1	rneqi 5929	. 2 ⊢ ran (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = ran (𝐴 ↾ 𝐵)
3		df-ima 5676	. 2 ⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = ran (𝐴 ↾ dom (𝐴 ↾ 𝐵))
4		df-ima 5676	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
5	2, 3, 4	3eqtr4i 2796	1 ⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵)

Syntax hints:   = wceq 1570  dom cdm 5663  ran crn 5664   ↾ cres 5665   “ cima 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is referenced by:  ssimaex  6968  fnwelem  8128  imafi  9276  r0weon  9997  limsupgle  15530  kqdisj  23870  isubgruhgr  48610