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Mirrors > Home > ILE Home > Th. List > opelxp | GIF version |
Description: Ordered pair membership in a cross product. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opelxp | ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp2 4495 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉) | |
2 | vex 2644 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | vex 2644 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opth2 4100 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ↔ (𝐴 = 𝑥 ∧ 𝐵 = 𝑦)) |
5 | eleq1 2162 | . . . . . . 7 ⊢ (𝐴 = 𝑥 → (𝐴 ∈ 𝐶 ↔ 𝑥 ∈ 𝐶)) | |
6 | eleq1 2162 | . . . . . . 7 ⊢ (𝐵 = 𝑦 → (𝐵 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷)) | |
7 | 5, 6 | bi2anan9 576 | . . . . . 6 ⊢ ((𝐴 = 𝑥 ∧ 𝐵 = 𝑦) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
8 | 4, 7 | sylbi 120 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
9 | 8 | biimprcd 159 | . . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))) |
10 | 9 | rexlimivv 2514 | . . 3 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
11 | eqid 2100 | . . . 4 ⊢ 〈𝐴, 𝐵〉 = 〈𝐴, 𝐵〉 | |
12 | opeq1 3652 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
13 | 12 | eqeq2d 2111 | . . . . 5 ⊢ (𝑥 = 𝐴 → (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ↔ 〈𝐴, 𝐵〉 = 〈𝐴, 𝑦〉)) |
14 | opeq2 3653 | . . . . . 6 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
15 | 14 | eqeq2d 2111 | . . . . 5 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝐵〉 = 〈𝐴, 𝑦〉 ↔ 〈𝐴, 𝐵〉 = 〈𝐴, 𝐵〉)) |
16 | 13, 15 | rspc2ev 2758 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 〈𝐴, 𝐵〉 = 〈𝐴, 𝐵〉) → ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉) |
17 | 11, 16 | mp3an3 1272 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉) |
18 | 10, 17 | impbii 125 | . 2 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
19 | 1, 18 | bitri 183 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1299 ∈ wcel 1448 ∃wrex 2376 〈cop 3477 × cxp 4475 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-opab 3930 df-xp 4483 |
This theorem is referenced by: brxp 4508 opelxpi 4509 opelxp1 4511 opelxp2 4512 opthprc 4528 elxp3 4531 opeliunxp 4532 optocl 4553 xpiindim 4614 opelres 4760 resiexg 4800 codir 4863 qfto 4864 xpmlem 4895 rnxpid 4909 ssrnres 4917 dfco2 4974 relssdmrn 4995 ressn 5015 opelf 5230 fnovex 5736 oprab4 5774 resoprab 5799 elmpocl 5900 fo1stresm 5990 fo2ndresm 5991 dfoprab4 6020 xporderlem 6058 f1od2 6062 brecop 6449 xpdom2 6654 djulclb 6855 djuss 6870 enq0enq 7140 enq0sym 7141 enq0tr 7143 nqnq0pi 7147 nnnq0lem1 7155 elinp 7183 genipv 7218 prsrlem1 7438 gt0srpr 7444 opelcn 7514 opelreal 7515 elreal2 7518 frecuzrdgrrn 10022 frec2uzrdg 10023 frecuzrdgrcl 10024 frecuzrdgsuc 10028 frecuzrdgrclt 10029 frecuzrdgsuctlem 10037 fisumcom2 11046 sqpweven 11645 2sqpwodd 11646 phimullem 11693 txuni2 12206 txcnp 12221 txcnmpt 12223 txdis1cn 12228 txlm 12229 xmeterval 12363 |
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