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Theorem imass2 4701
 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 4638 . . 3 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
2 rnss 4564 . . 3 ((𝐶𝐴) ⊆ (𝐶𝐵) → ran (𝐶𝐴) ⊆ ran (𝐶𝐵))
31, 2syl 14 . 2 (𝐴𝐵 → ran (𝐶𝐴) ⊆ ran (𝐶𝐵))
4 df-ima 4358 . 2 (𝐶𝐴) = ran (𝐶𝐴)
5 df-ima 4358 . 2 (𝐶𝐵) = ran (𝐶𝐵)
63, 4, 53sstr4g 2986 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ⊆ wss 2917  ran crn 4346   ↾ cres 4347   “ cima 4348 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by:  funimass1  4976  funimass2  4977  fvimacnv  5282  f1imass  5413  ecinxp  6181
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