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Mirrors > Home > MPE Home > Th. List > op1st | Structured version Visualization version GIF version |
Description: Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.) |
Ref | Expression |
---|---|
op1st.1 | ⊢ 𝐴 ∈ V |
op1st.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
op1st | ⊢ (1st ‘〈𝐴, 𝐵〉) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1stval 7680 | . 2 ⊢ (1st ‘〈𝐴, 𝐵〉) = ∪ dom {〈𝐴, 𝐵〉} | |
2 | op1st.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | op1st.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | op1sta 6075 | . 2 ⊢ ∪ dom {〈𝐴, 𝐵〉} = 𝐴 |
5 | 1, 4 | eqtri 2841 | 1 ⊢ (1st ‘〈𝐴, 𝐵〉) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 〈cop 4563 ∪ cuni 4830 dom cdm 5548 ‘cfv 6348 1st c1st 7676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-1st 7678 |
This theorem is referenced by: op1std 7688 op1stg 7690 1stval2 7695 fo1stres 7704 opreuopreu 7723 eloprabi 7750 algrflem 7808 xpmapenlem 8672 fseqenlem2 9439 archnq 10390 ruclem8 15578 idfu1st 17137 cofu1st 17141 xpccatid 17426 prf1st 17442 yonedalem21 17511 yonedalem22 17516 2ndcctbss 21991 upxp 22159 uptx 22161 cnheiborlem 23485 ovollb2lem 24016 ovolctb 24018 ovoliunlem2 24031 ovolshftlem1 24037 ovolscalem1 24041 ovolicc1 24044 addsqnreup 25946 2sqreuop 25965 2sqreuopnn 25966 2sqreuoplt 25967 2sqreuopltb 25968 2sqreuopnnlt 25969 2sqreuopnnltb 25970 ex-1st 28150 cnnvg 28382 cnnvs 28384 h2hva 28678 h2hsm 28679 hhssva 28961 hhsssm 28962 hhshsslem1 28971 eulerpartlemgvv 31533 eulerpartlemgh 31535 satfv0fvfmla0 32557 filnetlem3 33625 poimirlem17 34790 heiborlem8 34977 dvhvaddass 38113 dvhlveclem 38124 diblss 38186 pellexlem5 39308 pellex 39310 dvnprodlem1 42107 hoicvr 42707 hoicvrrex 42715 ovn0lem 42724 ovnhoilem1 42760 |
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