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Theorem abrexexd 32255
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 5947 . . 3 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
2 df-mpt 5225 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
32rneqi 5930 . . 3 ran (𝑥𝐴𝐵) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
4 df-rex 3065 . . . 4 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥𝐴𝑦 = 𝐵))
54abbii 2796 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
61, 3, 53eqtr4i 2764 . 2 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
7 abrexexd.1 . . 3 (𝜑𝐴 ∈ V)
8 funmpt 6580 . . . 4 Fun (𝑥𝐴𝐵)
9 eqid 2726 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
109dmmpt 6233 . . . . 5 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
11 abrexexd.0 . . . . . 6 𝑥𝐴
1211rabexgfGS 32248 . . . . 5 (𝐴 ∈ V → {𝑥𝐴𝐵 ∈ V} ∈ V)
1310, 12eqeltrid 2831 . . . 4 (𝐴 ∈ V → dom (𝑥𝐴𝐵) ∈ V)
14 funex 7216 . . . 4 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
158, 13, 14sylancr 586 . . 3 (𝐴 ∈ V → (𝑥𝐴𝐵) ∈ V)
16 rnexg 7892 . . 3 ((𝑥𝐴𝐵) ∈ V → ran (𝑥𝐴𝐵) ∈ V)
177, 15, 163syl 18 . 2 (𝜑 → ran (𝑥𝐴𝐵) ∈ V)
186, 17eqeltrrid 2832 1 (𝜑 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wex 1773  wcel 2098  {cab 2703  wnfc 2877  wrex 3064  {crab 3426  Vcvv 3468  {copab 5203  cmpt 5224  dom cdm 5669  ran crn 5670  Fun wfun 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545
This theorem is referenced by:  esumc  33579
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