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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 6060 | . 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 1331 ∈ wcel 1480 〈cop 3525 × cxp 4532 ‘cfv 5118 1st c1st 6029 2nd c2nd 6030 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fv 5126 df-1st 6031 df-2nd 6032 |
This theorem is referenced by: xpopth 6067 eqop 6068 2nd1st 6071 1st2nd 6072 xpmapenlem 6736 djuf1olem 6931 dfplpq2 7155 dfmpq2 7156 enqbreq2 7158 enqdc1 7163 preqlu 7273 prop 7276 elnp1st2nd 7277 cauappcvgprlemladd 7459 elreal2 7631 cnref1o 9433 frecuzrdgrrn 10174 frec2uzrdg 10175 frecuzrdgrcl 10176 frecuzrdgsuc 10180 frecuzrdgrclt 10181 frecuzrdgg 10182 frecuzrdgdomlem 10183 frecuzrdgfunlem 10185 frecuzrdgsuctlem 10189 seq3val 10224 seqvalcd 10225 eucalgval 11724 eucalginv 11726 eucalglt 11727 eucalg 11729 sqpweven 11842 2sqpwodd 11843 qnumdenbi 11859 tx1cn 12427 tx2cn 12428 txdis 12435 psmetxrge0 12490 xmetxpbl 12666 |
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