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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 5738 | . 2 ⊢ (A ∈ (B × 𝐶) ↔ (A = 〈(1st ‘A), (2nd ‘A)〉 ∧ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ 𝐶))) | |
2 | 1 | simplbi 259 | 1 ⊢ (A ∈ (B × 𝐶) → A = 〈(1st ‘A), (2nd ‘A)〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 〈cop 3370 × cxp 4286 ‘cfv 4845 1st c1st 5707 2nd c2nd 5708 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-iota 4810 df-fun 4847 df-fv 4853 df-1st 5709 df-2nd 5710 |
This theorem is referenced by: xpopth 5744 eqop 5745 2nd1st 5748 1st2nd 5749 dfplpq2 6338 dfmpq2 6339 enqbreq2 6341 enqdc1 6346 preqlu 6455 prop 6458 elnp1st2nd 6459 cauappcvgprlemladd 6630 elreal2 6728 cnref1o 8357 frecuzrdgrrn 8875 |
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