Theorem imadmres 6084

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 6082	. . 3 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵)
2	1	rneqi 5800	. 2 ⊢ ran (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = ran (𝐴 ↾ 𝐵)
3		df-ima 5561	. 2 ⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = ran (𝐴 ↾ dom (𝐴 ↾ 𝐵))
4		df-ima 5561	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
5	2, 3, 4	3eqtr4i 2852	1 ⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵)

Syntax hints:   = wceq 1531  dom cdm 5548  ran crn 5549   ↾ cres 5550   “ cima 5551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561

This theorem is referenced by:  ssimaex  6741  fnwelem  7817  imafi  8809  r0weon  9430  limsupgle  14826  kqdisj  22332