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Theorem imadmres 6190
Description: The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
Assertion
Ref Expression
imadmres (𝐴 “ dom (𝐴𝐵)) = (𝐴𝐵)

Proof of Theorem imadmres
StepHypRef Expression
1 resdmres 6188 . . 3 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
21rneqi 5896 . 2 ran (𝐴 ↾ dom (𝐴𝐵)) = ran (𝐴𝐵)
3 df-ima 5650 . 2 (𝐴 “ dom (𝐴𝐵)) = ran (𝐴 ↾ dom (𝐴𝐵))
4 df-ima 5650 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2771 1 (𝐴 “ dom (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  dom cdm 5637  ran crn 5638  cres 5639  cima 5640
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  ssimaex  6930  fnwelem  8067  imafiALT  9295  r0weon  9956  limsupgle  15368  kqdisj  23106
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