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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 4526 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑏∃𝑐(𝐴 = 〈𝑏, 𝑐〉 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶))) | |
2 | vex 2663 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
3 | vex 2663 | . . . . . . 7 ⊢ 𝑐 ∈ V | |
4 | 2, 3 | op2ndd 6015 | . . . . . 6 ⊢ (𝐴 = 〈𝑏, 𝑐〉 → (2nd ‘𝐴) = 𝑐) |
5 | 4 | eleq1d 2186 | . . . . 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 1852 | . 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 1316 ∃wex 1453 ∈ wcel 1465 〈cop 3500 × cxp 4507 ‘cfv 5093 2nd c2nd 6005 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fv 5101 df-2nd 6007 |
This theorem is referenced by: xpf1o 6706 xpmapenlem 6711 djuf1olem 6906 dfplpq2 7130 dfmpq2 7131 enqbreq2 7133 enqdc1 7138 mulpipq2 7147 preqlu 7248 elnp1st2nd 7252 cauappcvgprlemladd 7434 elreal2 7606 cnref1o 9408 frecuzrdgrrn 10149 frec2uzrdg 10150 frecuzrdgrcl 10151 frecuzrdgtcl 10153 frecuzrdgsuc 10155 frecuzrdgrclt 10156 frecuzrdgg 10157 frecuzrdgdomlem 10158 frecuzrdgfunlem 10160 frecuzrdgsuctlem 10164 seq3val 10199 seqvalcd 10200 fisumcom2 11175 eucalgval 11662 eucalginv 11664 eucalglt 11665 eucalgcvga 11666 eucalg 11667 sqpweven 11780 2sqpwodd 11781 ctiunctlemudc 11877 tx1cn 12365 txdis 12373 txhmeo 12415 xmetxp 12603 xmetxpbl 12604 xmettxlem 12605 xmettx 12606 |
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