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Theorem imadmres 4756
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 4755 . . 3 (A ↾ dom (AB)) = (AB)
21rneqi 4505 . 2 ran (A ↾ dom (AB)) = ran (AB)
3 df-ima 4301 . 2 (A “ dom (AB)) = ran (A ↾ dom (AB))
4 df-ima 4301 . 2 (AB) = ran (AB)
52, 3, 43eqtr4i 2067 1 (A “ dom (AB)) = (AB)
Colors of variables: wff set class
Syntax hints:   = wceq 1242  dom cdm 4288  ran crn 4289  cres 4290  cima 4291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by:  ssimaex  5177
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