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Theorem abrexexg 7657
Description: Existence of a class abstraction of existentially restricted sets. The class 𝐵 can be thought of as an expression in 𝑥 (which is typically a free variable in the class expression substituted for 𝐵) and the class abstraction appearing in the statement as the class of values 𝐵 as 𝑥 varies through 𝐴. If the "domain" 𝐴 is a set, then the abstraction is also a set. Therefore, this statement is a kind of Replacement. This can be seen by tracing back through the path mptexg 6975, funex 6973, fnex 6971, resfunexg 6969, and funimaexg 6428. See also abrexex2g 7660. There are partial converses under additional conditions, see for instance abnexg 7472. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
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 2824 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21rnmpt 5814 . 2 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
3 mptexg 6975 . . 3 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
4 rnexg 7609 . . 3 ((𝑥𝐴𝐵) ∈ V → ran (𝑥𝐴𝐵) ∈ V)
53, 4syl 17 . 2 (𝐴𝑉 → ran (𝑥𝐴𝐵) ∈ V)
62, 5eqeltrrid 2921 1 (𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {cab 2802  wrex 3134  Vcvv 3480  cmpt 5132  ran crn 5543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351
This theorem is referenced by:  abrexex  7658  iunexg  7659  qsexg  8351  wdomd  9042  cardiun  9408  rankcf  10197  sigaclci  31451  satf0suclem  32682  hbtlem1  39987  hbtlem7  39989  setpreimafvex  43830  fundcmpsurinj  43856  fundcmpsurbijinj  43857
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