MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1stval Unicode version

Theorem 1stval 6401
Description: The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
1stval

Proof of Theorem 1stval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sneq 3852 . . . . 5
21dmeqd 5116 . . . 4
32unieqd 4053 . . 3
4 df-1st 6399 . . 3
5 snex 4444 . . . . 5
65dmex 5175 . . . 4
76uniex 4746 . . 3
83, 4, 7fvmpt 5854 . 2
9 fvprc 5769 . . 3
10 snprc 3899 . . . . . . . 8
1110biimpi 188 . . . . . . 7
1211dmeqd 5116 . . . . . 6
13 dm0 5127 . . . . . 6
1412, 13syl6eq 2491 . . . . 5
1514unieqd 4053 . . . 4
16 uni0 4069 . . . 4
1715, 16syl6eq 2491 . . 3
189, 17eqtr4d 2478 . 2
198, 18pm2.61i 159 1
Colors of variables: wff set class
Syntax hints:  -.wn 3  =wceq 1654  e.wcel 1728   cvv 2965   c0 3616  {csn 3841  U.cuni 4043  domcdm 4919  `cfv 5501   c1st 6397
This theorem is referenced by:  1st0  6403  op1st  6405  1st2val  6422  elxp6  6428  mpt2xopxnop0  6516  1stnpr  24141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4364  ax-nul 4372  ax-pow 4416  ax-pr 4442  ax-un 4742
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3766  df-sn 3847  df-pr 3848  df-op 3850  df-uni 4044  df-br 4244  df-opab 4302  df-mpt 4303  df-id 4539  df-xp 4925  df-rel 4926  df-cnv 4927  df-co 4928  df-dm 4929  df-rn 4930  df-iota 5464  df-fun 5503  df-fv 5509  df-1st 6399
  Copyright terms: Public domain W3C validator