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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 5897. (Contributed by BJ, 25-Dec-2023.) |
Ref | Expression |
---|---|
bj-opelresdm | ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3903 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∩ (𝑋 × V)) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ (𝑋 × V))) | |
2 | opelxp1 5630 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑋 × V) → 𝐴 ∈ 𝑋) | |
3 | 1, 2 | simplbiim 505 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∩ (𝑋 × V)) → 𝐴 ∈ 𝑋) |
4 | df-res 5601 | . 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 3432 ∩ cin 3886 〈cop 4567 × cxp 5587 ↾ cres 5591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-opab 5137 df-xp 5595 df-res 5601 |
This theorem is referenced by: bj-brresdm 35317 |
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