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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssrel3 | Structured version Visualization version GIF version |
Description: Subclass relation in another form when the subclass is a relation. (Contributed by Peter Mazsa, 16-Feb-2019.) |
Ref | Expression |
---|---|
ssrel3 | ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrel 5639 | . 2 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
2 | df-br 5040 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | df-br 5040 | . . . 4 ⊢ (𝑥𝐵𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
4 | 2, 3 | imbi12i 354 | . . 3 ⊢ ((𝑥𝐴𝑦 → 𝑥𝐵𝑦) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵)) |
5 | 4 | 2albii 1828 | . 2 ⊢ (∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵)) |
6 | 1, 5 | bitr4di 292 | 1 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 ∈ wcel 2112 ⊆ wss 3853 〈cop 4533 class class class wbr 5039 Rel wrel 5541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-in 3860 df-ss 3870 df-br 5040 df-opab 5102 df-xp 5542 df-rel 5543 |
This theorem is referenced by: eqrel2 36121 inxpss 36133 inxpss2 36136 cocossss 36245 |
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