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Mirrors > Home > MPE Home > Th. List > relssdv | Structured version Visualization version GIF version |
Description: Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.) |
Ref | Expression |
---|---|
relssdv.1 | ⊢ (𝜑 → Rel 𝐴) |
relssdv.2 | ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)) |
Ref | Expression |
---|---|
relssdv | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relssdv.2 | . . 3 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)) | |
2 | 1 | alrimivv 1931 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)) |
3 | relssdv.1 | . . 3 ⊢ (𝜑 → Rel 𝐴) | |
4 | ssrel 5780 | . . 3 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))) |
6 | 2, 5 | mpbird 256 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1539 ∈ wcel 2106 ⊆ wss 3947 ⟨cop 4633 Rel wrel 5680 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-in 3954 df-ss 3964 df-opab 5210 df-xp 5681 df-rel 5682 |
This theorem is referenced by: relssres 6020 poirr2 6122 sofld 6183 relssdmrn 6264 relssdmrnOLD 6265 funcres2 17844 wunfunc 17845 wunfuncOLD 17846 fthres2 17879 pospo 18294 joindmss 18328 meetdmss 18342 clatl 18457 subrgdvds 20369 opsrtoslem2 21608 txcls 23099 txdis1cn 23130 txkgen 23147 qustgplem 23616 metustid 24054 metustexhalf 24056 ovoliunlem1 25010 dvres2 25420 cvmlift2lem12 34293 dib2dim 40102 dih2dimbALTN 40104 dihmeetlem1N 40149 dihglblem5apreN 40150 dihmeetlem13N 40178 dihjatcclem4 40280 |
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