Theorem resimass 45148

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 6031	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		imass1 6131	. 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ 𝐴 → ((𝐴 ↾ 𝐵) “ 𝐶) ⊆ (𝐴 “ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ↾ 𝐵) “ 𝐶) ⊆ (𝐴 “ 𝐶)

Syntax hints:   ⊆ wss 3976   ↾ cres 5702   “ cima 5703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by:  limsupres  45626  limsupresxr  45687  liminfresxr  45688