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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 7062 | . 2 ⊢ (2nd ‘〈𝐴, 𝐵〉) = ∪ ran {〈𝐴, 𝐵〉} | |
2 | op1st.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | op1st.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | op2nda 5538 | . 2 ⊢ ∪ ran {〈𝐴, 𝐵〉} = 𝐵 |
5 | 1, 4 | eqtri 2632 | 1 ⊢ (2nd ‘〈𝐴, 𝐵〉) = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 ∪ cuni 4372 ran crn 5039 ‘cfv 5804 2nd c2nd 7058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-2nd 7060 |
This theorem is referenced by: op2ndd 7070 op2ndg 7072 2ndval2 7077 fo2ndres 7084 eloprabi 7121 fo2ndf 7171 f1o2ndf1 7172 seqomlem1 7432 seqomlem2 7433 xpmapenlem 8012 fseqenlem2 8731 axdc4lem 9160 iunfo 9240 archnq 9681 om2uzrdg 12617 uzrdgsuci 12621 fsum2dlem 14343 fprod2dlem 14549 ruclem8 14805 ruclem11 14808 eucalglt 15136 idfu2nd 16360 idfucl 16364 cofu2nd 16368 cofucl 16371 xpccatid 16651 prf2nd 16668 curf2ndf 16710 yonedalem22 16741 gaid 17555 2ndcctbss 21068 upxp 21236 uptx 21238 txkgen 21265 cnheiborlem 22561 ovollb2lem 23063 ovolctb 23065 ovoliunlem2 23078 ovolshftlem1 23084 ovolscalem1 23088 ovolicc1 23091 iedgvalsnop 25717 wlknwwlknsur 26240 wlkiswwlksur 26247 clwlkfoclwwlk 26372 ex-2nd 26694 cnnvs 26919 cnnvnm 26920 h2hsm 27216 h2hnm 27217 hhsssm 27499 hhssnm 27500 aciunf1lem 28844 eulerpartlemgvv 29765 eulerpartlemgh 29767 msubff1 30707 msubvrs 30711 br2ndeq 30918 poimirlem17 32596 heiborlem7 32786 heiborlem8 32787 dvhvaddass 35404 dvhlveclem 35415 diblss 35477 pellexlem5 36415 pellex 36417 dvnprodlem1 38836 hoicvr 39438 hoicvrrex 39446 ovn0lem 39455 ovnhoilem1 39491 ovnlecvr2 39500 ovolval5lem2 39543 1wlkpwwlkf1ouspgr 41076 wlknwwlksnsur 41087 wlkwwlksur 41094 clwlksfoclwwlk 41270 |
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