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Theorem 1stval 6585
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 3920 . . . . 5
21dmeqd 5064 . . . 4
32unieqd 4127 . . 3
4 df-1st 6583 . . 3
5 snex 4556 . . . . 5
65dmex 6521 . . . 4
76uniex 6386 . . 3
83, 4, 7fvmpt 5790 . 2
9 fvprc 5702 . . 3
10 snprc 3971 . . . . . . . 8
1110biimpi 188 . . . . . . 7
1211dmeqd 5064 . . . . . 6
13 dm0 5075 . . . . . 6
1412, 13syl6eq 2537 . . . . 5
1514unieqd 4127 . . . 4
16 uni0 4144 . . . 4
1715, 16syl6eq 2537 . . 3
189, 17eqtr4d 2524 . 2
198, 18pm2.61i 159 1
Colors of variables: wff set class
Syntax hints:  -.wn 3  =wceq 1670  e.wcel 1732   cvv 3015   c0 3673  {csn 3909  U.cuni 4117  domcdm 4862  `cfv 5438   c1st 6581
This theorem is referenced by:  1stnpr  6587  1st0  6589  op1st  6591  1st2val  6608  elxp6  6614  mpt2xopxnop0  6698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1570  ax-4 1581  ax-5 1644  ax-6 1685  ax-7 1705  ax-8 1734  ax-9 1736  ax-10 1751  ax-11 1756  ax-12 1768  ax-13 1955  ax-ext 2470  ax-sep 4439  ax-nul 4447  ax-pow 4493  ax-pr 4554  ax-un 6382
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1338  df-ex 1566  df-nf 1569  df-sb 1677  df-eu 2317  df-mo 2318  df-clab 2476  df-cleq 2482  df-clel 2485  df-nfc 2614  df-ne 2654  df-ral 2764  df-rex 2765  df-rab 2768  df-v 3017  df-sbc 3225  df-dif 3368  df-un 3370  df-in 3372  df-ss 3379  df-nul 3674  df-if 3826  df-sn 3915  df-pr 3916  df-op 3918  df-uni 4118  df-br 4319  df-opab 4377  df-mpt 4378  df-id 4657  df-xp 4868  df-rel 4869  df-cnv 4870  df-co 4871  df-dm 4872  df-rn 4873  df-iota 5401  df-fun 5440  df-fv 5446  df-1st 6583
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