Theorem imadmres 6234

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 6232	. . 3 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵)
2	1	rneqi 5925	. 2 ⊢ ran (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = ran (𝐴 ↾ 𝐵)
3		df-ima 5672	. 2 ⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = ran (𝐴 ↾ dom (𝐴 ↾ 𝐵))
4		df-ima 5672	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
5	2, 3, 4	3eqtr4i 2795	1 ⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵)

This proof depends on syntax axioms:   = wceq 1570  dom cdm 5659  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:  ssimaex  6967  fnwelem  8133  imafi  9289  r0weon  10019  limsupgle  15568  kqdisj  23964  isubgruhgr  48792