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Mirrors > Home > MPE Home > Th. List > Mathboxes > idinxpssinxp | 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 34085. (Contributed by Peter Mazsa, 6-Mar-2019.) |
Ref | Expression |
---|---|
idinxpssinxp | ⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inxpss2 34085 | . 2 ⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 I 𝑦 → 𝑥𝑅𝑦)) | |
2 | ideqg 5273 | . . . . 5 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)) | |
3 | 2 | elv 33983 | . . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
4 | 3 | imbi1i 339 | . . 3 ⊢ ((𝑥 I 𝑦 → 𝑥𝑅𝑦) ↔ (𝑥 = 𝑦 → 𝑥𝑅𝑦)) |
5 | 4 | 2ralbii 2981 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 I 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦)) |
6 | 1, 5 | bitri 264 | 1 ⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∀wral 2912 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 I cid 5023 × cxp 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 |
This theorem is referenced by: idinxpssinxp4 34091 |
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