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Theorem imaeq1 4606
 Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
Assertion
Ref Expression
imaeq1 (A = B → (A𝐶) = (B𝐶))

Proof of Theorem imaeq1
StepHypRef Expression
1 reseq1 4549 . . 3 (A = B → (A𝐶) = (B𝐶))
21rneqd 4506 . 2 (A = B → ran (A𝐶) = ran (B𝐶))
3 df-ima 4301 . 2 (A𝐶) = ran (A𝐶)
4 df-ima 4301 . 2 (B𝐶) = ran (B𝐶)
52, 3, 43eqtr4g 2094 1 (A = B → (A𝐶) = (B𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242  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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301 This theorem is referenced by:  imaeq1i  4608  imaeq1d  4610  eceq2  6079
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