Theorem mptss 5877

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 5872	. 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶))
2		resss 5843	. 2 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶)
3	1, 2	eqsstrrdi 3970	1 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3881   ↦ cmpt 5110   ↾ cres 5521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093  df-mpt 5111  df-xp 5525  df-rel 5526  df-res 5531

This theorem is referenced by:  carsgclctunlem2  31687  sge0less  43031