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Mirrors > Home > ILE Home > Th. List > xp2nd | GIF version |
Description: Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
xp2nd | ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2nd ‘𝐴) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4556 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑏∃𝑐(𝐴 = 〈𝑏, 𝑐〉 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶))) | |
2 | vex 2689 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
3 | vex 2689 | . . . . . . 7 ⊢ 𝑐 ∈ V | |
4 | 2, 3 | op2ndd 6047 | . . . . . 6 ⊢ (𝐴 = 〈𝑏, 𝑐〉 → (2nd ‘𝐴) = 𝑐) |
5 | 4 | eleq1d 2208 | . . . . 5 ⊢ (𝐴 = 〈𝑏, 𝑐〉 → ((2nd ‘𝐴) ∈ 𝐶 ↔ 𝑐 ∈ 𝐶)) |
6 | 5 | biimpar 295 | . . . 4 ⊢ ((𝐴 = 〈𝑏, 𝑐〉 ∧ 𝑐 ∈ 𝐶) → (2nd ‘𝐴) ∈ 𝐶) |
7 | 6 | adantrl 469 | . . 3 ⊢ ((𝐴 = 〈𝑏, 𝑐〉 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (2nd ‘𝐴) ∈ 𝐶) |
8 | 7 | exlimivv 1868 | . 2 ⊢ (∃𝑏∃𝑐(𝐴 = 〈𝑏, 𝑐〉 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (2nd ‘𝐴) ∈ 𝐶) |
9 | 1, 8 | sylbi 120 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2nd ‘𝐴) ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 〈cop 3530 × cxp 4537 ‘cfv 5123 2nd c2nd 6037 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-2nd 6039 |
This theorem is referenced by: xpf1o 6738 xpmapenlem 6743 djuf1olem 6938 dfplpq2 7162 dfmpq2 7163 enqbreq2 7165 enqdc1 7170 mulpipq2 7179 preqlu 7280 elnp1st2nd 7284 cauappcvgprlemladd 7466 elreal2 7638 cnref1o 9440 frecuzrdgrrn 10181 frec2uzrdg 10182 frecuzrdgrcl 10183 frecuzrdgtcl 10185 frecuzrdgsuc 10187 frecuzrdgrclt 10188 frecuzrdgg 10189 frecuzrdgdomlem 10190 frecuzrdgfunlem 10192 frecuzrdgsuctlem 10196 seq3val 10231 seqvalcd 10232 fisumcom2 11207 eucalgval 11735 eucalginv 11737 eucalglt 11738 eucalgcvga 11739 eucalg 11740 sqpweven 11853 2sqpwodd 11854 ctiunctlemudc 11950 tx1cn 12438 txdis 12446 txhmeo 12488 xmetxp 12676 xmetxpbl 12677 xmettxlem 12678 xmettx 12679 |
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