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Theorem rnexg 4524
Description: The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.)
Assertion
Ref Expression
rnexg (A 𝑉 → ran A V)

Proof of Theorem rnexg
StepHypRef Expression
1 uniexg 4125 . 2 (A 𝑉 A V)
2 uniexg 4125 . 2 ( A V → A V)
3 ssun2 3084 . . . 4 ran A ⊆ (dom A ∪ ran A)
4 dmrnssfld 4522 . . . 4 (dom A ∪ ran A) ⊆ A
53, 4sstri 2931 . . 3 ran A A
6 ssexg 3870 . . 3 ((ran A A A V) → ran A V)
75, 6mpan 402 . 2 ( A V → ran A V)
81, 2, 73syl 17 1 (A 𝑉 → ran A V)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1374  Vcvv 2535  cun 2892  wss 2894   cuni 3554  dom cdm 4272  ran crn 4273
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-cnv 4280  df-dm 4282  df-rn 4283
This theorem is referenced by:  rnex  4526  imaexg  4607  xpexr2m  4689  elxp4  4735  elxp5  4736  cnvexg  4782  coexg  4789  fvexg  5119  cofunexg  5661  funrnex  5664  abrexexg  5668  2ndvalg  5693  tposexg  5795  iunon  5821
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