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Theorem fo2nd 6821
 Description: The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
fo2nd

Proof of Theorem fo2nd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4693 . . . . 5
21rnex 6734 . . . 4
32uniex 6596 . . 3
4 df-2nd 6801 . . 3
53, 4fnmpti 5714 . 2
64rnmpt 5253 . . 3
7 vex 3112 . . . . 5
8 opex 4716 . . . . . 6
97, 7op2nda 5498 . . . . . . 7
109eqcomi 2470 . . . . . 6
11 sneq 4039 . . . . . . . . . 10
1211rneqd 5235 . . . . . . . . 9
1312unieqd 4259 . . . . . . . 8
1413eqeq2d 2471 . . . . . . 7
1514rspcev 3210 . . . . . 6
168, 10, 15mp2an 672 . . . . 5
177, 162th 239 . . . 4
1817abbi2i 2590 . . 3
196, 18eqtr4i 2489 . 2
20 df-fo 5599 . 2
215, 19, 20mpbir2an 920 1
 Colors of variables: wff setvar class Syntax hints:  =wceq 1395  e.wcel 1818  {cab 2442  E.wrex 2808   cvv 3109  {csn 4029  <.cop 4035  U.cuni 4249  rancrn 5005  Fnwfn 5588  -onto->wfo 5591   c2nd 6799 This theorem is referenced by:  2ndcof  6829  df2nd2  6887  2ndconst  6889  iunfo  8935  cdaf  15377  2ndf1  15464  2ndf2  15465  2ndfcl  15467  gsum2dlem2  16998  gsum2dOLD  17000  upxp  20124  uptx  20126  cnmpt2nd  20170  uniiccdif  21987  xppreima  27487  xppreima2  27488  2ndpreima  27525  cnre2csqima  27893  br2ndeq  29205  filnetlem4  30199 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-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-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-fun 5595  df-fn 5596  df-fo 5599  df-2nd 6801
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