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Theorem abrexexg 5947
Description: Existence of a class abstraction of existentially restricted sets. 𝑥 is normally a free-variable parameter in 𝐵. The antecedent assures us that 𝐴 is a set. (Contributed by NM, 3-Nov-2003.)
Assertion
Ref Expression
abrexexg (𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem abrexexg
StepHypRef Expression
1 eqid 2100 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21rnmpt 4725 . 2 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
3 mptexg 5577 . . 3 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
4 rnexg 4740 . . 3 ((𝑥𝐴𝐵) ∈ V → ran (𝑥𝐴𝐵) ∈ V)
53, 4syl 14 . 2 (𝐴𝑉 → ran (𝑥𝐴𝐵) ∈ V)
62, 5syl5eqelr 2187 1 (𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  wcel 1448  {cab 2086  wrex 2376  Vcvv 2641  cmpt 3929  ran crn 4478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067
This theorem is referenced by:  iunexg  5948  qsexg  6415  shftfvalg  10431
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