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

Theorem fnasrn 5914
 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 5913 . 2
3 eqid 2438 . . . . 5
43rnmpt 5118 . . . 4
5 elsn 3831 . . . . . 6
65rexbii 2732 . . . . 5
76abbii 2550 . . . 4
84, 7eqtr4i 2461 . . 3
9 df-iun 4097 . . 3
108, 9eqtr4i 2461 . 2
112, 10eqtr4i 2461 1
 Colors of variables: wff set class Syntax hints:   wceq 1653   wcel 1726  cab 2424  wrex 2708  cvv 2958  csn 3816  cop 3819  ciun 4095   cmpt 4268   crn 4881 This theorem is referenced by:  resfunexg  5959  idref  5981  gruf  8688 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pr 4405 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-reu 2714  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-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463
 Copyright terms: Public domain W3C validator