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Theorem abrexexg 5798
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 2083 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21rnmpt 4631 . 2 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
3 mptexg 5440 . . 3 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
4 rnexg 4646 . . 3 ((𝑥𝐴𝐵) ∈ V → ran (𝑥𝐴𝐵) ∈ V)
53, 4syl 14 . 2 (𝐴𝑉 → ran (𝑥𝐴𝐵) ∈ V)
62, 5syl5eqelr 2170 1 (𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  {cab 2069  wrex 2354  Vcvv 2611  cmpt 3860  ran crn 4393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2613  df-sbc 2826  df-csb 2919  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961
This theorem is referenced by:  iunexg  5799  qsexg  6251  shftfvalg  9932
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