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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 6148 | . 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 1348 ∈ wcel 2141 〈cop 3586 × cxp 4609 ‘cfv 5198 1st c1st 6117 2nd c2nd 6118 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fv 5206 df-1st 6119 df-2nd 6120 |
This theorem is referenced by: xpopth 6155 eqop 6156 2nd1st 6159 1st2nd 6160 xpmapenlem 6827 djuf1olem 7030 dfplpq2 7316 dfmpq2 7317 enqbreq2 7319 enqdc1 7324 preqlu 7434 prop 7437 elnp1st2nd 7438 cauappcvgprlemladd 7620 elreal2 7792 cnref1o 9609 frecuzrdgrrn 10364 frec2uzrdg 10365 frecuzrdgrcl 10366 frecuzrdgsuc 10370 frecuzrdgrclt 10371 frecuzrdgg 10372 frecuzrdgdomlem 10373 frecuzrdgfunlem 10375 frecuzrdgsuctlem 10379 seq3val 10414 seqvalcd 10415 eucalgval 12008 eucalginv 12010 eucalglt 12011 eucalg 12013 sqpweven 12129 2sqpwodd 12130 qnumdenbi 12146 tx1cn 13063 tx2cn 13064 txdis 13071 psmetxrge0 13126 xmetxpbl 13302 |
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