|   | Mathbox for Peter Mazsa | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > inxpssidinxp | Structured version Visualization version GIF version | ||
| Description: Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 38317. (Contributed by Peter Mazsa, 4-Jul-2019.) | 
| Ref | Expression | 
|---|---|
| inxpssidinxp | ⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 = 𝑦)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | inxpss2 38317 | . 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 I 𝑦)) | |
| 2 | ideqg 5861 | . . . . 5 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)) | |
| 3 | 2 | elv 3484 | . . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) | 
| 4 | 3 | imbi2i 336 | . . 3 ⊢ ((𝑥𝑅𝑦 → 𝑥 I 𝑦) ↔ (𝑥𝑅𝑦 → 𝑥 = 𝑦)) | 
| 5 | 4 | 2ralbii 3127 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 I 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 = 𝑦)) | 
| 6 | 1, 5 | bitri 275 | 1 ⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 = 𝑦)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wral 3060 Vcvv 3479 ∩ cin 3949 ⊆ wss 3950 class class class wbr 5142 I cid 5576 × cxp 5682 | 
| 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 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 | 
| This theorem is referenced by: dfcnvrefrels3 38531 dfcnvrefrel3 38533 | 
| Copyright terms: Public domain | W3C validator |