Theorem mptss 5913

Description: Sufficient condition for inclusion among two functions in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
mptss	⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem mptss

Step	Hyp	Ref	Expression
1		resmpt 5908	. 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶))
2		resss 5881	. 2 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶)
3	1, 2	eqsstrrdi 4025	1 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3939   ↦ cmpt 5149   ↾ cres 5560

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-opab 5132  df-mpt 5150  df-xp 5564  df-rel 5565  df-res 5570

This theorem is referenced by:  carsgclctunlem2  31581  sge0less  42681