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| 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 38316. (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 38316 | . 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 I 𝑦)) | |
| 2 | ideqg 5862 | . . . . 5 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)) | |
| 3 | 2 | elv 3485 | . . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) | 
| 4 | 3 | imbi2i 336 | . . 3 ⊢ ((𝑥𝑅𝑦 → 𝑥 I 𝑦) ↔ (𝑥𝑅𝑦 → 𝑥 = 𝑦)) | 
| 5 | 4 | 2ralbii 3128 | . 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 3061 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 I cid 5577 × cxp 5683 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 | 
| This theorem is referenced by: dfcnvrefrels3 38530 dfcnvrefrel3 38532 | 
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