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Mirrors > Home > MPE Home > Th. List > op2ndg | Structured version Visualization version GIF version |
Description: Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
Ref | Expression |
---|---|
op2ndg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 4553 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | fveq2d 6357 | . . 3 ⊢ (𝑥 = 𝐴 → (2nd ‘〈𝑥, 𝑦〉) = (2nd ‘〈𝐴, 𝑦〉)) |
3 | 2 | eqeq1d 2762 | . 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘〈𝑥, 𝑦〉) = 𝑦 ↔ (2nd ‘〈𝐴, 𝑦〉) = 𝑦)) |
4 | opeq2 4554 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
5 | 4 | fveq2d 6357 | . . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘〈𝐴, 𝑦〉) = (2nd ‘〈𝐴, 𝐵〉)) |
6 | id 22 | . . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵) | |
7 | 5, 6 | eqeq12d 2775 | . 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘〈𝐴, 𝑦〉) = 𝑦 ↔ (2nd ‘〈𝐴, 𝐵〉) = 𝐵)) |
8 | vex 3343 | . . 3 ⊢ 𝑥 ∈ V | |
9 | vex 3343 | . . 3 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | op2nd 7343 | . 2 ⊢ (2nd ‘〈𝑥, 𝑦〉) = 𝑦 |
11 | 3, 7, 10 | vtocl2g 3410 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 〈cop 4327 ‘cfv 6049 2nd c2nd 7333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-iota 6012 df-fun 6051 df-fv 6057 df-2nd 7335 |
This theorem is referenced by: ot2ndg 7349 ot3rdg 7350 br2ndeqg 7357 2ndconst 7435 mpt2sn 7437 curry1 7438 xpmapenlem 8294 2ndinl 8964 2ndinr 8966 axdc4lem 9489 pinq 9961 addpipq 9971 mulpipq 9974 ordpipq 9976 swrdval 13636 ruclem1 15179 eucalg 15522 qnumdenbi 15674 setsstruct 16120 comffval 16580 oppccofval 16597 funcf2 16749 cofuval2 16768 resfval2 16774 resf2nd 16776 funcres 16777 isnat 16828 fucco 16843 homacd 16912 setcco 16954 catcco 16972 estrcco 16991 xpcco 17044 xpchom2 17047 xpcco2 17048 evlf2 17079 curfval 17084 curf1cl 17089 uncf1 17097 uncf2 17098 hof2fval 17116 yonedalem21 17134 yonedalem22 17139 mvmulfval 20570 imasdsf1olem 22399 ovolicc1 23504 ioombl1lem3 23548 ioombl1lem4 23549 brcgr 26000 opiedgfv 26107 fvtransport 32466 bj-elid3 33416 bj-finsumval0 33476 poimirlem17 33757 poimirlem24 33764 poimirlem27 33767 dvhopvadd 36902 dvhopvsca 36911 dvhopaddN 36923 dvhopspN 36924 etransclem44 41016 uspgrsprfo 42284 rngccoALTV 42516 ringccoALTV 42579 lmod1zr 42810 |
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