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Mirrors > Home > ILE Home > Th. List > 1st2nd2 | GIF version |
Description: Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.) |
Ref | Expression |
---|---|
1st2nd2 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp6 6117 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) | |
2 | 1 | simplbi 272 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 ∈ wcel 2128 〈cop 3563 × cxp 4584 ‘cfv 5170 1st c1st 6086 2nd c2nd 6087 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-iota 5135 df-fun 5172 df-fv 5178 df-1st 6088 df-2nd 6089 |
This theorem is referenced by: xpopth 6124 eqop 6125 2nd1st 6128 1st2nd 6129 xpmapenlem 6794 djuf1olem 6997 dfplpq2 7274 dfmpq2 7275 enqbreq2 7277 enqdc1 7282 preqlu 7392 prop 7395 elnp1st2nd 7396 cauappcvgprlemladd 7578 elreal2 7750 cnref1o 9559 frecuzrdgrrn 10307 frec2uzrdg 10308 frecuzrdgrcl 10309 frecuzrdgsuc 10313 frecuzrdgrclt 10314 frecuzrdgg 10315 frecuzrdgdomlem 10316 frecuzrdgfunlem 10318 frecuzrdgsuctlem 10322 seq3val 10357 seqvalcd 10358 eucalgval 11930 eucalginv 11932 eucalglt 11933 eucalg 11935 sqpweven 12049 2sqpwodd 12050 qnumdenbi 12066 tx1cn 12669 tx2cn 12670 txdis 12677 psmetxrge0 12732 xmetxpbl 12908 |
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