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Theorem 1stval 6540
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 3905 . . . . 5
21dmeqd 5046 . . . 4
32unieqd 4111 . . 3
4 df-1st 6538 . . 3
5 snex 4540 . . . . 5
65dmex 6476 . . . 4
76uniex 6342 . . 3
83, 4, 7fvmpt 5767 . 2
9 fvprc 5679 . . 3
10 snprc 3956 . . . . . . . 8
1110biimpi 188 . . . . . . 7
1211dmeqd 5046 . . . . . 6
13 dm0 5057 . . . . . 6
1412, 13syl6eq 2529 . . . . 5
1514unieqd 4111 . . . 4
16 uni0 4128 . . . 4
1715, 16syl6eq 2529 . . 3
189, 17eqtr4d 2516 . 2
198, 18pm2.61i 159 1
Colors of variables: wff set class
Syntax hints:  -.wn 3  =wceq 1662  e.wcel 1724   cvv 3006   c0 3660  {csn 3894  U.cuni 4101  domcdm 4844  `cfv 5417   c1st 6536
This theorem is referenced by:  1stnpr  6542  1st0  6544  op1st  6546  1st2val  6563  elxp6  6569  mpt2xopxnop0  6649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1562  ax-4 1573  ax-5 1636  ax-6 1677  ax-7 1697  ax-8 1726  ax-9 1728  ax-10 1743  ax-11 1748  ax-12 1760  ax-13 1947  ax-ext 2462  ax-sep 4423  ax-nul 4431  ax-pow 4477  ax-pr 4538  ax-un 6338
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1337  df-ex 1558  df-nf 1561  df-sb 1669  df-eu 2309  df-mo 2310  df-clab 2468  df-cleq 2474  df-clel 2477  df-nfc 2606  df-ne 2646  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3008  df-sbc 3213  df-dif 3356  df-un 3358  df-in 3360  df-ss 3367  df-nul 3661  df-if 3813  df-sn 3900  df-pr 3901  df-op 3903  df-uni 4102  df-br 4303  df-opab 4361  df-mpt 4362  df-id 4639  df-xp 4850  df-rel 4851  df-cnv 4852  df-co 4853  df-dm 4854  df-rn 4855  df-iota 5380  df-fun 5419  df-fv 5425  df-1st 6538
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