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Mirrors > Home > MPE Home > Th. List > op2nd | Structured version Visualization version Unicode version |
Description: Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.) |
Ref | Expression |
---|---|
op1st.1 | |
op1st.2 |
Ref | Expression |
---|---|
op2nd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2ndval 7171 | . 2 | |
2 | op1st.1 | . . 3 | |
3 | op1st.2 | . . 3 | |
4 | 2, 3 | op2nda 5620 | . 2 |
5 | 1, 4 | eqtri 2644 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 cvv 3200 csn 4177 cop 4183 cuni 4436 crn 5115 cfv 5888 c2nd 7167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-2nd 7169 |
This theorem is referenced by: op2ndd 7179 op2ndg 7181 2ndval2 7186 fo2ndres 7193 eloprabi 7232 fo2ndf 7284 f1o2ndf1 7285 seqomlem1 7545 seqomlem2 7546 xpmapenlem 8127 fseqenlem2 8848 axdc4lem 9277 iunfo 9361 archnq 9802 om2uzrdg 12755 uzrdgsuci 12759 fsum2dlem 14501 fprod2dlem 14710 ruclem8 14966 ruclem11 14969 eucalglt 15298 idfu2nd 16537 idfucl 16541 cofu2nd 16545 cofucl 16548 xpccatid 16828 prf2nd 16845 curf2ndf 16887 yonedalem22 16918 gaid 17732 2ndcctbss 21258 upxp 21426 uptx 21428 txkgen 21455 cnheiborlem 22753 ovollb2lem 23256 ovolctb 23258 ovoliunlem2 23271 ovolshftlem1 23277 ovolscalem1 23281 ovolicc1 23284 wlkpwwlkf1ouspgr 26765 wlknwwlksnsur 26776 wlkwwlksur 26783 clwlksfoclwwlk 26963 ex-2nd 27302 cnnvs 27535 cnnvnm 27536 h2hsm 27832 h2hnm 27833 hhsssm 28115 hhssnm 28116 aciunf1lem 29462 eulerpartlemgvv 30438 eulerpartlemgh 30440 msubff1 31453 msubvrs 31457 br2ndeq 31671 poimirlem17 33426 heiborlem7 33616 heiborlem8 33617 dvhvaddass 36386 dvhlveclem 36397 diblss 36459 pellexlem5 37397 pellex 37399 dvnprodlem1 40161 hoicvr 40762 hoicvrrex 40770 ovn0lem 40779 ovnhoilem1 40815 ovnlecvr2 40824 ovolval5lem2 40867 |
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