| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 4873 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
| 2 | 1 | fveq2d 6910 | . . 3 ⊢ (𝑥 = 𝐴 → (1st ‘〈𝑥, 𝑦〉) = (1st ‘〈𝐴, 𝑦〉)) |
| 3 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
| 4 | 2, 3 | eqeq12d 2753 | . 2 ⊢ (𝑥 = 𝐴 → ((1st ‘〈𝑥, 𝑦〉) = 𝑥 ↔ (1st ‘〈𝐴, 𝑦〉) = 𝐴)) |
| 5 | opeq2 4874 | . . 3 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
| 6 | 5 | fveqeq2d 6914 | . 2 ⊢ (𝑦 = 𝐵 → ((1st ‘〈𝐴, 𝑦〉) = 𝐴 ↔ (1st ‘〈𝐴, 𝐵〉) = 𝐴)) |
| 7 | vex 3484 | . . 3 ⊢ 𝑥 ∈ V | |
| 8 | vex 3484 | . . 3 ⊢ 𝑦 ∈ V | |
| 9 | 7, 8 | op1st 8022 | . 2 ⊢ (1st ‘〈𝑥, 𝑦〉) = 𝑥 |
| 10 | 4, 6, 9 | vtocl2g 3574 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 ‘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: ot1stg 8028 ot2ndg 8029 br1steqg 8036 1stconst 8125 mposn 8128 curry2 8132 opco1 8148 mpoxopn0yelv 8238 mpoxopoveq 8244 xpmapenlem 9184 1stinl 9967 1stinr 9969 fpwwe 10686 addpipq 10977 mulpipq 10980 ordpipq 10982 swrdval 14681 ruclem1 16267 qnumdenbi 16781 setsstruct 17213 oppccofval 17759 funcf2 17913 cofuval2 17932 resfval2 17938 resf1st 17939 isnat 17995 fucco 18010 homadm 18085 setcco 18128 estrcco 18174 xpcco 18228 xpchom2 18231 xpcco2 18232 evlf2 18263 curfval 18268 curf1cl 18273 uncf1 18281 uncf2 18282 diag11 18288 diag12 18289 diag2 18290 hof2fval 18300 yonedalem21 18318 yonedalem22 18323 mvmulfval 22548 imasdsf1olem 24383 ovolicc1 25551 ioombl1lem3 25595 ioombl1lem4 25596 addsqnreup 27487 addsval 27995 mulsval 28135 brcgr 28915 opvtxfv 29021 fgreu 32682 fsuppcurry2 32737 erlbrd 33267 rlocaddval 33272 rlocmulval 33273 fracerl 33308 sategoelfvb 35424 prv1n 35436 fvtransport 36033 bj-inftyexpiinv 37209 bj-finsumval0 37286 poimirlem17 37644 poimirlem24 37651 poimirlem27 37654 rngoablo2 37916 dvhopvadd 41095 dvhopvsca 41104 dvhopaddN 41116 dvhopspN 41117 etransclem44 46293 ovnsubaddlem1 46585 ovnlecvr2 46625 ovolval5lem2 46668 rngccoALTV 48187 ringccoALTV 48221 upfval3 48935 swapf1val 48973 fucofval 49014 fuco111 49025 fuco21 49031 fucoid 49043 precoffunc 49066 |
| Copyright terms: Public domain | W3C validator |