![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-opelresdm | Structured version Visualization version GIF version |
Description: If an ordered pair is in a restricted binary relation, then its first component is an element of the restricting class. See also opelres 6006. (Contributed by BJ, 25-Dec-2023.) |
Ref | Expression |
---|---|
bj-opelresdm | ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3979 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∩ (𝑋 × V)) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ (𝑋 × V))) | |
2 | opelxp1 5731 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑋 × V) → 𝐴 ∈ 𝑋) | |
3 | 1, 2 | simplbiim 504 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∩ (𝑋 × V)) → 𝐴 ∈ 𝑋) |
4 | df-res 5701 | . 2 ⊢ (𝑅 ↾ 𝑋) = (𝑅 ∩ (𝑋 × V)) | |
5 | 3, 4 | eleq2s 2857 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 〈cop 4637 × cxp 5687 ↾ cres 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-xp 5695 df-res 5701 |
This theorem is referenced by: bj-brresdm 37129 |
Copyright terms: Public domain | W3C validator |