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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 7430 | . 2 ⊢ (1st ‘〈𝐴, 𝐵〉) = ∪ dom {〈𝐴, 𝐵〉} | |
2 | op1st.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | op1st.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | op1sta 5859 | . 2 ⊢ ∪ dom {〈𝐴, 𝐵〉} = 𝐴 |
5 | 1, 4 | eqtri 2849 | 1 ⊢ (1st ‘〈𝐴, 𝐵〉) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 Vcvv 3414 {csn 4397 〈cop 4403 ∪ cuni 4658 dom cdm 5342 ‘cfv 6123 1st c1st 7426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-iota 6086 df-fun 6125 df-fv 6131 df-1st 7428 |
This theorem is referenced by: op1std 7438 op1stg 7440 1stval2 7445 fo1stres 7454 eloprabi 7495 algrflem 7550 xpmapenlem 8396 fseqenlem2 9161 archnq 10117 ruclem8 15340 idfu1st 16891 cofu1st 16895 xpccatid 17181 prf1st 17197 yonedalem21 17266 yonedalem22 17271 2ndcctbss 21629 upxp 21797 uptx 21799 cnheiborlem 23123 ovollb2lem 23654 ovolctb 23656 ovoliunlem2 23669 ovolshftlem1 23675 ovolscalem1 23679 ovolicc1 23682 wlknwwlksnsurOLD 27190 wlkwwlksurOLD 27198 clwlksfoclwwlkOLD 27432 ex-1st 27848 cnnvg 28077 cnnvs 28079 h2hva 28375 h2hsm 28376 hhssva 28658 hhsssm 28659 hhshsslem1 28668 eulerpartlemgvv 30972 eulerpartlemgh 30974 filnetlem3 32902 poimirlem17 33963 heiborlem8 34152 dvhvaddass 37165 dvhlveclem 37176 diblss 37238 pellexlem5 38234 pellex 38236 dvnprodlem1 40949 hoicvr 41549 hoicvrrex 41557 ovn0lem 41566 ovnhoilem1 41602 |
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