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Mirrors > Home > MPE Home > Th. List > mptss | Structured version Visualization version GIF version |
Description: Sufficient condition for inclusion 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 5686 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
2 | resss 5658 | . 2 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶) | |
3 | 1, 2 | syl6eqssr 3881 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3798 ↦ cmpt 4952 ↾ cres 5344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-opab 4936 df-mpt 4953 df-xp 5348 df-rel 5349 df-res 5354 |
This theorem is referenced by: carsgclctunlem2 30926 sge0less 41400 |
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