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Theorem f2ndres 6372
 Description: Mapping of a restriction of the (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
f2ndres

Proof of Theorem f2ndres
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2961 . . . . . . . 8
2 vex 2961 . . . . . . . 8
31, 2op2nda 5357 . . . . . . 7
43eleq1i 2501 . . . . . 6
54biimpri 199 . . . . 5
65adantl 454 . . . 4
76rgen2 2804 . . 3
8 sneq 3827 . . . . . . 7
98rneqd 5100 . . . . . 6
109unieqd 4028 . . . . 5
1110eleq1d 2504 . . . 4
1211ralxp 5019 . . 3
137, 12mpbir 202 . 2
14 df-2nd 6353 . . . . 5
1514reseq1i 5145 . . . 4
16 ssv 3370 . . . . 5
17 resmpt 5194 . . . . 5
1816, 17ax-mp 5 . . . 4
1915, 18eqtri 2458 . . 3
2019fmpt 5893 . 2
2113, 20mpbi 201 1
 Colors of variables: wff set class Syntax hints:   wceq 1653   wcel 1726  wral 2707  cvv 2958   wss 3322  csn 3816  cop 3819  cuni 4017   cmpt 4269   cxp 4879   crn 4882   cres 4883  wf 5453  c2nd 6351 This theorem is referenced by:  fo2ndres  6374  2ndcof  6378  fparlem2  6450  f2ndf  6455  eucalgcvga  13082  2ndfcl  14300  gaid  15081  tx2cn  17647  txkgen  17689  xpinpreima  24309  xpinpreima2  24310  2ndmbfm  24616  filnetlem4  26424  hausgraph  27522 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 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-2nd 6353
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