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Theorem 2ndval 6352
 Description: The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
2ndval

Proof of Theorem 2ndval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sneq 3825 . . . . 5
21rneqd 5097 . . . 4
32unieqd 4026 . . 3
4 df-2nd 6350 . . 3
5 snex 4405 . . . . 5
65rnex 5133 . . . 4
76uniex 4705 . . 3
83, 4, 7fvmpt 5806 . 2
9 fvprc 5722 . . 3
10 snprc 3871 . . . . . . . 8
1110biimpi 187 . . . . . . 7
1211rneqd 5097 . . . . . 6
13 rn0 5127 . . . . . 6
1412, 13syl6eq 2484 . . . . 5
1514unieqd 4026 . . . 4
16 uni0 4042 . . . 4
1715, 16syl6eq 2484 . . 3
189, 17eqtr4d 2471 . 2
198, 18pm2.61i 158 1
 Colors of variables: wff set class Syntax hints:   wn 3   wceq 1652   wcel 1725  cvv 2956  c0 3628  csn 3814  cuni 4015   crn 4879  cfv 5454  c2nd 6348 This theorem is referenced by:  2nd0  6354  op2nd  6356  2nd2val  6373  elxp6  6378  2ndnpr  24094 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-2nd 6350
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