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Theorem imass2 4644
 Description: Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
imass2 (AB → (𝐶A) ⊆ (𝐶B))

Proof of Theorem imass2
StepHypRef Expression
1 ssres2 4581 . . 3 (AB → (𝐶A) ⊆ (𝐶B))
2 rnss 4507 . . 3 ((𝐶A) ⊆ (𝐶B) → ran (𝐶A) ⊆ ran (𝐶B))
31, 2syl 14 . 2 (AB → ran (𝐶A) ⊆ ran (𝐶B))
4 df-ima 4301 . 2 (𝐶A) = ran (𝐶A)
5 df-ima 4301 . 2 (𝐶B) = ran (𝐶B)
63, 4, 53sstr4g 2980 1 (AB → (𝐶A) ⊆ (𝐶B))
 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-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301 This theorem is referenced by:  funimass1  4919  funimass2  4920  fvimacnv  5225  f1imass  5356  ecinxp  6117
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