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Theorem fnasrn 5341
 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 5340 . 2
3 eqid 2040 . . . . 5
43rnmpt 4582 . . . 4
5 velsn 3392 . . . . . 6
65rexbii 2331 . . . . 5
76abbii 2153 . . . 4
84, 7eqtr4i 2063 . . 3
9 df-iun 3659 . . 3
108, 9eqtr4i 2063 . 2
112, 10eqtr4i 2063 1
 Colors of variables: wff set class Syntax hints:   wceq 1243   wcel 1393  cab 2026  wrex 2307  cvv 2557  csn 3375  cop 3378  ciun 3657   cmpt 3818   crn 4346 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909 This theorem is referenced by:  idref  5396
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