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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 3550 | . . 3 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
2 | 1 | eleq1d 2106 | . 2 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 ∈ (𝐶 ↾ 𝐷) ↔ 〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷))) |
3 | 1 | eleq1d 2106 | . . 3 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝑦〉 ∈ 𝐶 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶)) |
4 | 3 | anbi1d 438 | . 2 ⊢ (𝑦 = 𝐵 → ((〈𝐴, 𝑦〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))) |
5 | vex 2560 | . . 3 ⊢ 𝑦 ∈ V | |
6 | 5 | opelres 4617 | . 2 ⊢ (〈𝐴, 𝑦〉 ∈ (𝐶 ↾ 𝐷) ↔ (〈𝐴, 𝑦〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
7 | 2, 4, 6 | vtoclbg 2614 | 1 ⊢ (𝐵 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ (𝐶 ↾ 𝐷) ↔ (〈𝐴, 𝐵〉 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 〈cop 3378 ↾ cres 4347 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-xp 4351 df-res 4357 |
This theorem is referenced by: brresg 4620 opelresi 4623 issref 4707 |
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