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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 5987. (Contributed by BJ, 25-Dec-2023.) |
Ref | Expression |
---|---|
bj-opelresdm | ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3964 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∩ (𝑋 × V)) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ (𝑋 × V))) | |
2 | opelxp1 5718 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑋 × V) → 𝐴 ∈ 𝑋) | |
3 | 1, 2 | simplbiim 504 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∩ (𝑋 × V)) → 𝐴 ∈ 𝑋) |
4 | df-res 5688 | . 2 ⊢ (𝑅 ↾ 𝑋) = (𝑅 ∩ (𝑋 × V)) | |
5 | 3, 4 | eleq2s 2850 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3473 ∩ cin 3947 〈cop 4634 × cxp 5674 ↾ cres 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-opab 5211 df-xp 5682 df-res 5688 |
This theorem is referenced by: bj-brresdm 36493 |
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