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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 4402 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | fveq2d 6195 | . . 3 ⊢ (𝑥 = 𝐴 → (2nd ‘〈𝑥, 𝑦〉) = (2nd ‘〈𝐴, 𝑦〉)) |
3 | 2 | eqeq1d 2624 | . 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘〈𝑥, 𝑦〉) = 𝑦 ↔ (2nd ‘〈𝐴, 𝑦〉) = 𝑦)) |
4 | opeq2 4403 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
5 | 4 | fveq2d 6195 | . . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘〈𝐴, 𝑦〉) = (2nd ‘〈𝐴, 𝐵〉)) |
6 | id 22 | . . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵) | |
7 | 5, 6 | eqeq12d 2637 | . 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘〈𝐴, 𝑦〉) = 𝑦 ↔ (2nd ‘〈𝐴, 𝐵〉) = 𝐵)) |
8 | vex 3203 | . . 3 ⊢ 𝑥 ∈ V | |
9 | vex 3203 | . . 3 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | op2nd 7177 | . 2 ⊢ (2nd ‘〈𝑥, 𝑦〉) = 𝑦 |
11 | 3, 7, 10 | vtocl2g 3270 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 〈cop 4183 ‘cfv 5888 2nd c2nd 7167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-2nd 7169 |
This theorem is referenced by: ot2ndg 7183 ot3rdg 7184 2ndconst 7266 mpt2sn 7268 curry1 7269 xpmapenlem 8127 axdc4lem 9277 pinq 9749 addpipq 9759 mulpipq 9762 ordpipq 9764 swrdval 13417 ruclem1 14960 eucalg 15300 qnumdenbi 15452 setsstruct 15898 comffval 16359 oppccofval 16376 funcf2 16528 cofuval2 16547 resfval2 16553 resf2nd 16555 funcres 16556 isnat 16607 fucco 16622 homacd 16691 setcco 16733 catcco 16751 estrcco 16770 xpcco 16823 xpchom2 16826 xpcco2 16827 evlf2 16858 curfval 16863 curf1cl 16868 uncf1 16876 uncf2 16877 hof2fval 16895 yonedalem21 16913 yonedalem22 16918 mvmulfval 20348 imasdsf1olem 22178 ovolicc1 23284 ioombl1lem3 23328 ioombl1lem4 23329 brcgr 25780 opiedgfv 25887 br2ndeqg 31673 fvtransport 32139 bj-elid3 33087 bj-finsumval0 33147 poimirlem17 33426 poimirlem24 33433 poimirlem27 33436 dvhopvadd 36382 dvhopvsca 36391 dvhopaddN 36403 dvhopspN 36404 etransclem44 40495 uspgrsprfo 41756 rngccoALTV 41988 ringccoALTV 42051 lmod1zr 42282 |
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