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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 5824. (Contributed by BJ, 25-Dec-2023.) |
Ref | Expression |
---|---|
bj-opelresdm | ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3897 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∩ (𝑋 × V)) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ (𝑋 × V))) | |
2 | opelxp1 5560 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑋 × V) → 𝐴 ∈ 𝑋) | |
3 | 1, 2 | simplbiim 508 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∩ (𝑋 × V)) → 𝐴 ∈ 𝑋) |
4 | df-res 5531 | . 2 ⊢ (𝑅 ↾ 𝑋) = (𝑅 ∩ (𝑋 × V)) | |
5 | 3, 4 | eleq2s 2908 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 〈cop 4531 × cxp 5517 ↾ cres 5521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-xp 5525 df-res 5531 |
This theorem is referenced by: bj-brresdm 34561 |
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