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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 4362 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑏∃𝑐(𝐴 = 〈𝑏, 𝑐〉 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶))) | |
2 | vex 2560 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
3 | vex 2560 | . . . . . . 7 ⊢ 𝑐 ∈ V | |
4 | 2, 3 | op2ndd 5776 | . . . . . 6 ⊢ (𝐴 = 〈𝑏, 𝑐〉 → (2nd ‘𝐴) = 𝑐) |
5 | 4 | eleq1d 2106 | . . . . 5 ⊢ (𝐴 = 〈𝑏, 𝑐〉 → ((2nd ‘𝐴) ∈ 𝐶 ↔ 𝑐 ∈ 𝐶)) |
6 | 5 | biimpar 281 | . . . 4 ⊢ ((𝐴 = 〈𝑏, 𝑐〉 ∧ 𝑐 ∈ 𝐶) → (2nd ‘𝐴) ∈ 𝐶) |
7 | 6 | adantrl 447 | . . 3 ⊢ ((𝐴 = 〈𝑏, 𝑐〉 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (2nd ‘𝐴) ∈ 𝐶) |
8 | 7 | exlimivv 1776 | . 2 ⊢ (∃𝑏∃𝑐(𝐴 = 〈𝑏, 𝑐〉 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (2nd ‘𝐴) ∈ 𝐶) |
9 | 1, 8 | sylbi 114 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2nd ‘𝐴) ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∃wex 1381 ∈ wcel 1393 〈cop 3378 × cxp 4343 ‘cfv 4902 2nd c2nd 5766 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fv 4910 df-2nd 5768 |
This theorem is referenced by: dfplpq2 6452 dfmpq2 6453 enqbreq2 6455 enqdc1 6460 mulpipq2 6469 preqlu 6570 elnp1st2nd 6574 cauappcvgprlemladd 6756 elreal2 6907 cnref1o 8582 frecuzrdgrrn 9194 frec2uzrdg 9195 frecuzrdgfn 9198 frecuzrdgcl 9199 frecuzrdgsuc 9201 |
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