Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnasrn Unicode version

Theorem fnasrn 6077
 Description: A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfmpt.1
Assertion
Ref Expression
fnasrn

Proof of Theorem fnasrn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfmpt.1 . . 3
21dfmpt 6076 . 2
3 eqid 2457 . . . . 5
43rnmpt 5253 . . . 4
5 elsn 4043 . . . . . 6
65rexbii 2959 . . . . 5
76abbii 2591 . . . 4
84, 7eqtr4i 2489 . . 3
9 df-iun 4332 . . 3
108, 9eqtr4i 2489 . 2
112, 10eqtr4i 2489 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_ciun 4330  e.cmpt 4510  rancrn 5005 This theorem is referenced by:  resfunexg  6137  idref  6153  gruf  9210 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-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 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-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  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-iun 4332  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-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600
 Copyright terms: Public domain W3C validator