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Mirrors > Home > ILE Home > Th. List > opelresg | GIF version |
Description: Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
opelresg | ⊢ (𝐵 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq2 3779 | . . 3 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
2 | 1 | eleq1d 2246 | . 2 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 ∈ (𝐶 ↾ 𝐷) ↔ 〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷))) |
3 | 1 | eleq1d 2246 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 ∈ 𝐶 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶)) |
4 | 3 | anbi1d 465 | . 2 ⊢ (𝑦 = 𝐵 → ((〈𝐴, 𝑦〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))) |
5 | vex 2740 | . . 3 ⊢ 𝑦 ∈ V | |
6 | 5 | opelres 4912 | . 2 ⊢ (〈𝐴, 𝑦〉 ∈ (𝐶 ↾ 𝐷) ↔ (〈𝐴, 𝑦〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
7 | 2, 4, 6 | vtoclbg 2798 | 1 ⊢ (𝐵 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 〈cop 3595 ↾ cres 4628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-opab 4065 df-xp 4632 df-res 4638 |
This theorem is referenced by: brresg 4915 opelresi 4918 issref 5011 |
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