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| 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 5791 | . 2 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
| 2 | df-br 5143 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
| 3 | df-br 5143 | . . . 4 ⊢ (𝑥𝐵𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
| 4 | 2, 3 | imbi12i 350 | . . 3 ⊢ ((𝑥𝐴𝑦 → 𝑥𝐵𝑦) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵)) | 
| 5 | 4 | 2albii 1819 | . 2 ⊢ (∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵)) | 
| 6 | 1, 5 | bitr4di 289 | 1 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1537 ∈ wcel 2107 ⊆ wss 3950 〈cop 4631 class class class wbr 5142 Rel wrel 5689 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 | 
| This theorem is referenced by: cotrg 6126 cotrgOLD 6127 cnvsym 6131 cnvsymOLD 6132 eqrel2 38301 inxpss 38313 inxpss2 38317 cnvref5 38353 cocossss 38438 | 
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