Theorem imass2d 45784

Description: Subset theorem for image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
imass2d.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
imass2d	⊢ (𝜑 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵))

Proof of Theorem imass2d

Step	Hyp	Ref	Expression
1		imass2d.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		imass2 6081	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵))

Syntax hints:   → wi 4   ⊆ wss 3899   “ cima 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-br 5095  df-opab 5157  df-xp 5646  df-cnv 5648  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653

This theorem is referenced by:  liminflelimsuplem  46297