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Theorem abrexexd 29625
Description: Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.)
Hypotheses
Ref Expression
abrexexd.0 𝑥𝐴
abrexexd.1 (𝜑𝐴 ∈ V)
Assertion
Ref Expression
abrexexd (𝜑 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem abrexexd
StepHypRef Expression
1 rnopab 5513 . . 3 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
2 df-mpt 4870 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
32rneqi 5495 . . 3 ran (𝑥𝐴𝐵) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
4 df-rex 3044 . . . 4 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥𝐴𝑦 = 𝐵))
54abbii 2865 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
61, 3, 53eqtr4i 2780 . 2 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
7 abrexexd.1 . . 3 (𝜑𝐴 ∈ V)
8 funmpt 6075 . . . 4 Fun (𝑥𝐴𝐵)
9 eqid 2748 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
109dmmpt 5779 . . . . 5 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
11 abrexexd.0 . . . . . 6 𝑥𝐴
1211rabexgfGS 29618 . . . . 5 (𝐴 ∈ V → {𝑥𝐴𝐵 ∈ V} ∈ V)
1310, 12syl5eqel 2831 . . . 4 (𝐴 ∈ V → dom (𝑥𝐴𝐵) ∈ V)
14 funex 6634 . . . 4 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
158, 13, 14sylancr 698 . . 3 (𝐴 ∈ V → (𝑥𝐴𝐵) ∈ V)
16 rnexg 7251 . . 3 ((𝑥𝐴𝐵) ∈ V → ran (𝑥𝐴𝐵) ∈ V)
177, 15, 163syl 18 . 2 (𝜑 → ran (𝑥𝐴𝐵) ∈ V)
186, 17syl5eqelr 2832 1 (𝜑 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wex 1841  wcel 2127  {cab 2734  wnfc 2877  wrex 3039  {crab 3042  Vcvv 3328  {copab 4852  cmpt 4869  dom cdm 5254  ran crn 5255  Fun wfun 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045
This theorem is referenced by:  esumc  30393
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