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Mirrors > Home > MPE Home > Th. List > op1stg | Structured version Visualization version GIF version |
Description: Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
Ref | Expression |
---|---|
op1stg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 4540 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | fveq2d 6337 | . . 3 ⊢ (𝑥 = 𝐴 → (1st ‘〈𝑥, 𝑦〉) = (1st ‘〈𝐴, 𝑦〉)) |
3 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
4 | 2, 3 | eqeq12d 2786 | . 2 ⊢ (𝑥 = 𝐴 → ((1st ‘〈𝑥, 𝑦〉) = 𝑥 ↔ (1st ‘〈𝐴, 𝑦〉) = 𝐴)) |
5 | opeq2 4541 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
6 | 5 | fveq2d 6337 | . . 3 ⊢ (𝑦 = 𝐵 → (1st ‘〈𝐴, 𝑦〉) = (1st ‘〈𝐴, 𝐵〉)) |
7 | 6 | eqeq1d 2773 | . 2 ⊢ (𝑦 = 𝐵 → ((1st ‘〈𝐴, 𝑦〉) = 𝐴 ↔ (1st ‘〈𝐴, 𝐵〉) = 𝐴)) |
8 | vex 3354 | . . 3 ⊢ 𝑥 ∈ V | |
9 | vex 3354 | . . 3 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | op1st 7327 | . 2 ⊢ (1st ‘〈𝑥, 𝑦〉) = 𝑥 |
11 | 4, 7, 10 | vtocl2g 3421 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 〈cop 4323 ‘cfv 6030 1st c1st 7317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-iota 5993 df-fun 6032 df-fv 6038 df-1st 7319 |
This theorem is referenced by: ot1stg 7333 ot2ndg 7334 br1steqg 7341 1stconst 7420 mpt2sn 7423 curry2 7427 mpt2xopn0yelv 7495 mpt2xopoveq 7501 xpmapenlem 8287 1stinl 8957 1stinr 8959 fpwwe 9674 addpipq 9965 mulpipq 9968 ordpipq 9970 swrdval 13625 ruclem1 15166 qnumdenbi 15659 setsstruct 16105 oppccofval 16583 funcf2 16735 cofuval2 16754 resfval2 16760 resf1st 16761 isnat 16814 fucco 16829 homadm 16897 setcco 16940 estrcco 16977 xpcco 17031 xpchom2 17034 xpcco2 17035 evlf2 17066 curfval 17071 curf1cl 17076 uncf1 17084 uncf2 17085 diag11 17091 diag12 17092 diag2 17093 hof2fval 17103 yonedalem21 17121 yonedalem22 17126 mvmulfval 20566 imasdsf1olem 22398 ovolicc1 23504 ioombl1lem3 23548 ioombl1lem4 23549 brcgr 26001 opvtxfv 26105 fgreu 29811 fvtransport 32476 bj-elid3 33423 bj-inftyexpiinv 33431 bj-finsumval0 33483 poimirlem17 33758 poimirlem24 33765 poimirlem27 33768 rngoablo2 34038 dvhopvadd 36901 dvhopvsca 36910 dvhopaddN 36922 dvhopspN 36923 etransclem44 41007 ovnsubaddlem1 41299 ovnlecvr2 41339 ovolval5lem2 41382 rngccoALTV 42511 ringccoALTV 42574 |
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