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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 7716 | . 2 ⊢ (1st ‘〈𝐴, 𝐵〉) = ∪ dom {〈𝐴, 𝐵〉} | |
2 | op1st.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | op1st.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | op1sta 6057 | . 2 ⊢ ∪ dom {〈𝐴, 𝐵〉} = 𝐴 |
5 | 1, 4 | eqtri 2761 | 1 ⊢ (1st ‘〈𝐴, 𝐵〉) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 Vcvv 3398 {csn 4516 〈cop 4522 ∪ cuni 4796 dom cdm 5525 ‘cfv 6339 1st c1st 7712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-iota 6297 df-fun 6341 df-fv 6347 df-1st 7714 |
This theorem is referenced by: op1std 7724 op1stg 7726 1stval2 7731 fo1stres 7740 opreuopreu 7759 eloprabi 7786 algrflem 7845 xpmapenlem 8734 fseqenlem2 9525 archnq 10480 ruclem8 15682 idfu1st 17254 cofu1st 17258 xpccatid 17554 prf1st 17570 yonedalem21 17639 yonedalem22 17644 2ndcctbss 22206 upxp 22374 uptx 22376 cnheiborlem 23706 ovollb2lem 24240 ovolctb 24242 ovoliunlem2 24255 ovolshftlem1 24261 ovolscalem1 24265 ovolicc1 24268 addsqnreup 26179 2sqreuop 26198 2sqreuopnn 26199 2sqreuoplt 26200 2sqreuopltb 26201 2sqreuopnnlt 26202 2sqreuopnnltb 26203 ex-1st 28381 cnnvg 28613 cnnvs 28615 h2hva 28909 h2hsm 28910 hhssva 29192 hhsssm 29193 hhshsslem1 29202 gsumhashmul 30893 eulerpartlemgvv 31913 eulerpartlemgh 31915 satfv0fvfmla0 32946 ot21std 33254 filnetlem3 34207 poimirlem17 35417 heiborlem8 35599 dvhvaddass 38734 dvhlveclem 38745 diblss 38807 pellexlem5 40227 pellex 40229 dvnprodlem1 43029 hoicvr 43628 hoicvrrex 43636 ovn0lem 43645 ovnhoilem1 43681 |
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