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Theorem rnexg 4540
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 4141 . 2 (A 𝑉 A V)
2 uniexg 4141 . 2 ( A V → A V)
3 ssun2 3101 . . . 4 ran A ⊆ (dom A ∪ ran A)
4 dmrnssfld 4538 . . . 4 (dom A ∪ ran A) ⊆ A
53, 4sstri 2948 . . 3 ran A A
6 ssexg 3887 . . 3 ((ran A A A V) → ran A V)
75, 6mpan 400 . 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 1390  Vcvv 2551  cun 2909  wss 2911   cuni 3571  dom cdm 4288  ran crn 4289
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  rnex  4542  imaexg  4623  xpexr2m  4705  elxp4  4751  elxp5  4752  cnvexg  4798  coexg  4805  fvexg  5137  cofunexg  5680  funrnex  5683  abrexexg  5687  2ndvalg  5712  tposexg  5814  iunon  5840  fopwdom  6246
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