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Theorem 2ndval 6803
 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 4039 . . . . 5
21rneqd 5235 . . . 4
32unieqd 4259 . . 3
4 df-2nd 6801 . . 3
5 snex 4693 . . . . 5
65rnex 6734 . . . 4
76uniex 6596 . . 3
83, 4, 7fvmpt 5956 . 2
9 fvprc 5865 . . 3
10 snprc 4093 . . . . . . . 8
1110biimpi 194 . . . . . . 7
1211rneqd 5235 . . . . . 6
13 rn0 5259 . . . . . 6
1412, 13syl6eq 2514 . . . . 5
1514unieqd 4259 . . . 4
16 uni0 4276 . . . 4
1715, 16syl6eq 2514 . . 3
189, 17eqtr4d 2501 . 2
198, 18pm2.61i 164 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  =wceq 1395  e.wcel 1818   cvv 3109   c0 3784  {csn 4029  U.cuni 4249  rancrn 5005  `cfv 5593   c2nd 6799 This theorem is referenced by:  2ndnpr  6805  2nd0  6807  op2nd  6809  2nd2val  6827  elxp6  6832 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fv 5601  df-2nd 6801
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