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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 8016 | . 2 ⊢ (1st ‘〈𝐴, 𝐵〉) = ∪ dom {〈𝐴, 𝐵〉} | |
| 2 | op1st.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 3 | op1st.2 | . . 3 ⊢ 𝐵 ∈ V | |
| 4 | 2, 3 | op1sta 6245 | . 2 ⊢ ∪ dom {〈𝐴, 𝐵〉} = 𝐴 |
| 5 | 1, 4 | eqtri 2765 | 1 ⊢ (1st ‘〈𝐴, 𝐵〉) = 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 ∪ cuni 4907 dom cdm 5685 ‘cfv 6561 1st c1st 8012 |
| 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-1st 8014 |
| This theorem is referenced by: op1std 8024 op1stg 8026 1stval2 8031 fo1stres 8040 opreuopreu 8059 eloprabi 8088 xpmapenlem 9184 fseqenlem2 10065 archnq 11020 ruclem8 16273 idfu1st 17924 cofu1st 17928 xpccatid 18233 prf1st 18249 yonedalem21 18318 yonedalem22 18323 2ndcctbss 23463 upxp 23631 uptx 23633 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 precsexlem1 28231 precsexlem4 28234 ex-1st 30463 cnnvg 30697 cnnvs 30699 h2hva 30993 h2hsm 30994 hhssva 31276 hhsssm 31277 hhshsslem1 31286 gsumhashmul 33064 rlocf1 33277 fracfld 33310 eulerpartlemgvv 34378 eulerpartlemgh 34380 satfv0fvfmla0 35418 filnetlem3 36381 poimirlem17 37644 heiborlem8 37825 dvhvaddass 41099 dvhlveclem 41110 diblss 41172 aks6d1c3 42124 pellexlem5 42844 pellex 42846 dvnprodlem1 45961 hoicvr 46563 hoicvrrex 46571 ovn0lem 46580 ovnhoilem1 46616 gpgedgvtx0 48019 gpgedgvtx1 48020 gpg3kgrtriex 48045 swapf1vala 48972 swapf2f1oaALT 48984 swapfcoa 48987 fuco21 49031 fucof21 49042 thincciso 49102 |
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