Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mptss | Structured version Visualization version GIF version |
Description: Sufficient condition for inclusion among two functions in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
mptss | ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resmpt 5898 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
2 | resss 5871 | . 2 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶) | |
3 | 1, 2 | eqsstrrdi 4019 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3933 ↦ cmpt 5137 ↾ cres 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-mpt 5138 df-xp 5554 df-rel 5555 df-res 5560 |
This theorem is referenced by: carsgclctunlem2 31476 sge0less 42551 |
Copyright terms: Public domain | W3C validator |