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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 4831 | . . 3 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩) | |
2 | 1 | fveqeq2d 6851 | . 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)) |
3 | opeq2 4832 | . . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩) | |
4 | 3 | fveq2d 6847 | . . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) |
5 | id 22 | . . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵) | |
6 | 4, 5 | eqeq12d 2753 | . 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)) |
7 | vex 3450 | . . 3 ⊢ 𝑥 ∈ V | |
8 | vex 3450 | . . 3 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | op2nd 7931 | . 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 |
10 | 2, 6, 9 | vtocl2g 3532 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⟨cop 4593 ‘cfv 6497 2nd c2nd 7921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fv 6505 df-2nd 7923 |
This theorem is referenced by: ot2ndg 7937 ot3rdg 7938 br2ndeqg 7945 2ndconst 8034 mposn 8036 curry1 8037 opco2 8057 xpmapenlem 9089 2ndinl 9865 2ndinr 9867 axdc4lem 10392 pinq 10864 addpipq 10874 mulpipq 10877 ordpipq 10879 swrdval 14532 ruclem1 16114 eucalg 16464 qnumdenbi 16620 setsstruct 17049 comffval 17580 oppccofval 17598 funcf2 17755 cofuval2 17774 resfval2 17780 resf2nd 17782 funcres 17783 isnat 17835 fucco 17852 homacd 17928 setcco 17970 catcco 17992 estrcco 18018 xpcco 18072 xpchom2 18075 xpcco2 18076 evlf2 18108 curfval 18113 curf1cl 18118 uncf1 18126 uncf2 18127 hof2fval 18145 yonedalem21 18163 yonedalem22 18168 mvmulfval 21894 imasdsf1olem 23729 ovolicc1 24883 ioombl1lem3 24927 ioombl1lem4 24928 addsqnreup 26794 addsval 27277 brcgr 27852 opiedgfv 27961 fsuppcurry1 31645 sategoelfvb 34016 prv1n 34028 mulsval 34411 fvtransport 34620 bj-finsumval0 35759 poimirlem17 36098 poimirlem24 36105 poimirlem27 36108 dvhopvadd 39559 dvhopvsca 39568 dvhopaddN 39580 dvhopspN 39581 etransclem44 44526 uspgrsprfo 46057 rngccoALTV 46293 ringccoALTV 46356 lmod1zr 46581 |
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