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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 4831 | . . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩) | |
2 | 1 | fveq2d 6847 | . . 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 4832 | . . 3 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩) | |
6 | 5 | fveqeq2d 6851 | . 2 ⊢ (𝑦 = 𝐵 → ((1st ‘⟨𝐴, 𝑦⟩) = 𝐴 ↔ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)) |
7 | vex 3450 | . . 3 ⊢ 𝑥 ∈ V | |
8 | vex 3450 | . . 3 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | op1st 7930 | . 2 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥 |
10 | 4, 6, 9 | vtocl2g 3532 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⟨cop 4593 ‘cfv 6497 1st c1st 7920 |
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-1st 7922 |
This theorem is referenced by: ot1stg 7936 ot2ndg 7937 br1steqg 7944 1stconst 8033 mposn 8036 curry2 8040 opco1 8056 mpoxopn0yelv 8145 mpoxopoveq 8151 xpmapenlem 9089 1stinl 9864 1stinr 9866 fpwwe 10583 addpipq 10874 mulpipq 10877 ordpipq 10879 swrdval 14532 ruclem1 16114 qnumdenbi 16620 setsstruct 17049 oppccofval 17598 funcf2 17755 cofuval2 17774 resfval2 17780 resf1st 17781 isnat 17835 fucco 17852 homadm 17927 setcco 17970 estrcco 18018 xpcco 18072 xpchom2 18075 xpcco2 18076 evlf2 18108 curfval 18113 curf1cl 18118 uncf1 18126 uncf2 18127 diag11 18133 diag12 18134 diag2 18135 hof2fval 18145 yonedalem21 18163 yonedalem22 18168 mvmulfval 21894 imasdsf1olem 23729 ovolicc1 24883 ioombl1lem3 24927 ioombl1lem4 24928 addsqnreup 26794 addsval 27277 brcgr 27852 opvtxfv 27958 fgreu 31591 fsuppcurry2 31646 sategoelfvb 34016 prv1n 34028 mulsval 34411 fvtransport 34620 bj-inftyexpiinv 35682 bj-finsumval0 35759 poimirlem17 36098 poimirlem24 36105 poimirlem27 36108 rngoablo2 36371 dvhopvadd 39559 dvhopvsca 39568 dvhopaddN 39580 dvhopspN 39581 etransclem44 44526 ovnsubaddlem1 44818 ovnlecvr2 44858 ovolval5lem2 44901 rngccoALTV 46293 ringccoALTV 46356 |
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