Theorem imadmres 6137

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 6135	. . 3 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵)
2	1	rneqi 5846	. 2 ⊢ ran (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = ran (𝐴 ↾ 𝐵)
3		df-ima 5602	. 2 ⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = ran (𝐴 ↾ dom (𝐴 ↾ 𝐵))
4		df-ima 5602	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
5	2, 3, 4	3eqtr4i 2776	1 ⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵)

Syntax hints:   = wceq 1539  dom cdm 5589  ran crn 5590   ↾ cres 5591   “ cima 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602

This theorem is referenced by:  ssimaex  6853  fnwelem  7972  imafiALT  9112  r0weon  9768  limsupgle  15186  kqdisj  22883