Theorem resimass 45246

Description: The image of a restriction is a subset of the original image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Assertion

Ref	Expression
resimass	⊢ ((𝐴 ↾ 𝐵) “ 𝐶) ⊆ (𝐴 “ 𝐶)

Proof of Theorem resimass

Step	Hyp	Ref	Expression
1		resss 6019	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		imass1 6119	. 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ 𝐴 → ((𝐴 ↾ 𝐵) “ 𝐶) ⊆ (𝐴 “ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ↾ 𝐵) “ 𝐶) ⊆ (𝐴 “ 𝐶)

Syntax hints:   ⊆ wss 3951   ↾ cres 5687   “ cima 5688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698

This theorem is referenced by:  limsupres  45720  limsupresxr  45781  liminfresxr  45782