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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 1929 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵)) |
3 | relssdv.1 | . . 3 ⊢ (𝜑 → Rel 𝐴) | |
4 | ssrel 5657 | . . 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 259 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∈ wcel 2114 ⊆ wss 3936 〈cop 4573 Rel wrel 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3943 df-ss 3952 df-opab 5129 df-xp 5561 df-rel 5562 |
This theorem is referenced by: relssres 5893 poirr2 5984 sofld 6044 relssdmrn 6121 funcres2 17168 wunfunc 17169 fthres2 17202 pospo 17583 joindmss 17617 meetdmss 17631 clatl 17726 subrgdvds 19549 opsrtoslem2 20265 txcls 22212 txdis1cn 22243 txkgen 22260 qustgplem 22729 metustid 23164 metustexhalf 23166 ovoliunlem1 24103 dvres2 24510 cvmlift2lem12 32561 dib2dim 38394 dih2dimbALTN 38396 dihmeetlem1N 38441 dihglblem5apreN 38442 dihmeetlem13N 38470 dihjatcclem4 38572 |
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