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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 4616 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉) | |
2 | vex 2724 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | vex 2724 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opth2 4212 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ↔ (𝐴 = 𝑥 ∧ 𝐵 = 𝑦)) |
5 | eleq1 2227 | . . . . . . 7 ⊢ (𝐴 = 𝑥 → (𝐴 ∈ 𝐶 ↔ 𝑥 ∈ 𝐶)) | |
6 | eleq1 2227 | . . . . . . 7 ⊢ (𝐵 = 𝑦 → (𝐵 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷)) | |
7 | 5, 6 | bi2anan9 596 | . . . . . 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 2587 | . . 3 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
11 | eqid 2164 | . . . 4 ⊢ 〈𝐴, 𝐵〉 = 〈𝐴, 𝐵〉 | |
12 | opeq1 3752 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
13 | 12 | eqeq2d 2176 | . . . . 5 ⊢ (𝑥 = 𝐴 → (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ↔ 〈𝐴, 𝐵〉 = 〈𝐴, 𝑦〉)) |
14 | opeq2 3753 | . . . . . 6 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
15 | 14 | eqeq2d 2176 | . . . . 5 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝐵〉 = 〈𝐴, 𝑦〉 ↔ 〈𝐴, 𝐵〉 = 〈𝐴, 𝐵〉)) |
16 | 13, 15 | rspc2ev 2840 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 〈𝐴, 𝐵〉 = 〈𝐴, 𝐵〉) → ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉) |
17 | 11, 16 | mp3an3 1315 | . . 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 1342 ∈ wcel 2135 ∃wrex 2443 〈cop 3573 × cxp 4596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 df-xp 4604 |
This theorem is referenced by: brxp 4629 opelxpi 4630 opelxp1 4632 opelxp2 4633 opthprc 4649 elxp3 4652 opeliunxp 4653 optocl 4674 xpiindim 4735 opelres 4883 resiexg 4923 codir 4986 qfto 4987 xpmlem 5018 rnxpid 5032 ssrnres 5040 dfco2 5097 relssdmrn 5118 ressn 5138 opelf 5353 fnovex 5866 oprab4 5904 resoprab 5929 elmpocl 6030 fo1stresm 6121 fo2ndresm 6122 dfoprab4 6152 xporderlem 6190 f1od2 6194 brecop 6582 xpdom2 6788 djulclb 7011 djuss 7026 enq0enq 7363 enq0sym 7364 enq0tr 7366 nqnq0pi 7370 nnnq0lem1 7378 elinp 7406 genipv 7441 prsrlem1 7674 gt0srpr 7680 opelcn 7758 opelreal 7759 elreal2 7762 frecuzrdgrrn 10333 frec2uzrdg 10334 frecuzrdgrcl 10335 frecuzrdgsuc 10339 frecuzrdgrclt 10340 frecuzrdgsuctlem 10348 fisumcom2 11365 fprodcom2fi 11553 sqpweven 12086 2sqpwodd 12087 phimullem 12136 txuni2 12803 txcnp 12818 txcnmpt 12820 txdis1cn 12825 txlm 12826 xmeterval 12982 limccnp2lem 13192 limccnp2cntop 13193 |
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