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| Mirrors > Home > MPE Home > Th. List > op2nd | Structured version Visualization version GIF version | ||
| Description: Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.) |
| Ref | Expression |
|---|---|
| op1st.1 | ⊢ 𝐴 ∈ V |
| op1st.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| op2nd | ⊢ (2nd ‘〈𝐴, 𝐵〉) = 𝐵 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2ndval 8017 | . 2 ⊢ (2nd ‘〈𝐴, 𝐵〉) = ∪ ran {〈𝐴, 𝐵〉} | |
| 2 | op1st.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 3 | op1st.2 | . . 3 ⊢ 𝐵 ∈ V | |
| 4 | 2, 3 | op2nda 6248 | . 2 ⊢ ∪ ran {〈𝐴, 𝐵〉} = 𝐵 |
| 5 | 1, 4 | eqtri 2765 | 1 ⊢ (2nd ‘〈𝐴, 𝐵〉) = 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 ∪ cuni 4907 ran crn 5686 ‘cfv 6561 2nd c2nd 8013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-2nd 8015 |
| This theorem is referenced by: op2ndd 8025 op2ndg 8027 2ndval2 8032 fo2ndres 8041 opreuopreu 8059 eloprabi 8088 fo2ndf 8146 f1o2ndf1 8147 seqomlem1 8490 seqomlem2 8491 xpmapenlem 9184 fseqenlem2 10065 axdc4lem 10495 iunfo 10579 archnq 11020 om2uzrdg 13997 uzrdgsuci 14001 fsum2dlem 15806 fprod2dlem 16016 ruclem8 16273 ruclem11 16276 eucalglt 16622 idfu2nd 17922 idfucl 17926 cofu2nd 17930 cofucl 17933 xpccatid 18233 prf2nd 18250 curf2ndf 18292 yonedalem22 18323 gaid 19317 2ndcctbss 23463 upxp 23631 uptx 23633 txkgen 23660 cnheiborlem 24986 ovollb2lem 25523 ovolctb 25525 ovoliunlem2 25538 ovolshftlem1 25544 ovolscalem1 25548 ovolicc1 25551 addsqnreup 27487 2sqreuop 27506 2sqreuopnn 27507 2sqreuoplt 27508 2sqreuopltb 27509 2sqreuopnnlt 27510 2sqreuopnnltb 27511 precsexlem2 28232 precsexlem5 28235 om2noseqrdg 28310 noseqrdgsuc 28314 wlkswwlksf1o 29899 clwlkclwwlkfo 30028 ex-2nd 30464 cnnvs 30699 cnnvnm 30700 h2hsm 30994 h2hnm 30995 hhsssm 31277 hhssnm 31278 2ndimaxp 32656 2ndresdju 32659 aciunf1lem 32672 gsumpart 33060 rlocf1 33277 fracfld 33310 eulerpartlemgvv 34378 eulerpartlemgh 34380 satfv0fvfmla0 35418 sategoelfvb 35424 prv1n 35436 msubff1 35561 msubvrs 35565 poimirlem17 37644 heiborlem7 37824 heiborlem8 37825 dvhvaddass 41099 dvhlveclem 41110 diblss 41172 aks6d1c3 42124 pellexlem5 42844 pellex 42846 dvnprodlem1 45961 hoicvr 46563 hoicvrrex 46571 ovn0lem 46580 ovnhoilem1 46616 ovnlecvr2 46625 ovolval5lem2 46668 gpg3kgrtriex 48045 swapf2fvala 48970 swapf2f1oaALT 48984 swapfcoa 48987 fuco21 49031 fucof21 49042 |
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