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Theorem imass1 4643
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
Assertion
Ref Expression
imass1 (AB → (A𝐶) ⊆ (B𝐶))

Proof of Theorem imass1
StepHypRef Expression
1 ssres 4580 . . 3 (AB → (A𝐶) ⊆ (B𝐶))
2 rnss 4507 . . 3 ((A𝐶) ⊆ (B𝐶) → ran (A𝐶) ⊆ ran (B𝐶))
31, 2syl 14 . 2 (AB → ran (A𝐶) ⊆ ran (B𝐶))
4 df-ima 4301 . 2 (A𝐶) = ran (A𝐶)
5 df-ima 4301 . 2 (B𝐶) = ran (B𝐶)
63, 4, 53sstr4g 2980 1 (AB → (A𝐶) ⊆ (B𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wss 2911  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: (None)
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