Theorem imass2d 44519

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 6094	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵))

Syntax hints:   → wi 4   ⊆ wss 3943   “ cima 5672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682

This theorem is referenced by:  liminflelimsuplem  45044