Theorem mptss 5944

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 5939	. 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶))
2		resss 5910	. 2 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶)
3	1, 2	eqsstrrdi 3976	1 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3887   ↦ cmpt 5157   ↾ cres 5587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-opab 5137  df-mpt 5158  df-xp 5591  df-rel 5592  df-res 5597

This theorem is referenced by:  tdeglem4  25212  carsgclctunlem2  32272  sge0less  43889