Theorem resimass 46455

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 6002	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		imass1 6065	. 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ 𝐴 → ((𝐴 ↾ 𝐵) “ 𝐶) ⊆ (𝐴 “ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ↾ 𝐵) “ 𝐶) ⊆ (𝐴 “ 𝐶)

This proof depends on syntax axioms:   ⊆ wss 3900   ↾ cres 5657   “ cima 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668

This theorem is used by:  limsupres  46918  limsupresxr  46979  liminfresxr  46980