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Theorem rnexg 4625
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 (𝐴𝑉 → ran 𝐴 ∈ V)

Proof of Theorem rnexg
StepHypRef Expression
1 uniexg 4203 . 2 (𝐴𝑉 𝐴 ∈ V)
2 uniexg 4203 . 2 ( 𝐴 ∈ V → 𝐴 ∈ V)
3 ssun2 3135 . . . 4 ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
4 dmrnssfld 4623 . . . 4 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
53, 4sstri 2982 . . 3 ran 𝐴 𝐴
6 ssexg 3924 . . 3 ((ran 𝐴 𝐴 𝐴 ∈ V) → ran 𝐴 ∈ V)
75, 6mpan 408 . 2 ( 𝐴 ∈ V → ran 𝐴 ∈ V)
81, 2, 73syl 17 1 (𝐴𝑉 → ran 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1409  Vcvv 2574  cun 2943  wss 2945   cuni 3608  dom cdm 4373  ran crn 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-cnv 4381  df-dm 4383  df-rn 4384
This theorem is referenced by:  rnex  4627  imaexg  4708  xpexr2m  4790  elxp4  4836  elxp5  4837  cnvexg  4883  coexg  4890  fvexg  5222  cofunexg  5766  funrnex  5769  abrexexg  5773  2ndvalg  5798  tposexg  5904  iunon  5930  fopwdom  6341  shftfvalg  9647  ovshftex  9648
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