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Theorem abrexex 5771
Description: Existence of a class abstraction of existentially restricted sets. 𝑥 is normally a free-variable parameter in the class expression substituted for 𝐵, which can be thought of as 𝐵(𝑥). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5413, funex 5411, fnex 5410, resfunexg 5409, and funimaexg 5010. See also abrexex2 5778. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
abrexex.1 𝐴 ∈ V
Assertion
Ref Expression
abrexex {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem abrexex
StepHypRef Expression
1 eqid 2056 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21rnmpt 4609 . 2 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
3 abrexex.1 . . . 4 𝐴 ∈ V
43mptex 5414 . . 3 (𝑥𝐴𝐵) ∈ V
54rnex 4626 . 2 ran (𝑥𝐴𝐵) ∈ V
62, 5eqeltrri 2127 1 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  {cab 2042  wrex 2324  Vcvv 2574  cmpt 3845  ran crn 4373
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-coll 3899  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
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-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937
This theorem is referenced by:  ab2rexex  5785  shftfval  9643
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