Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  abrexexd Structured version   Visualization version   GIF version

Theorem abrexexd 30854
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 5863 . . 3 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
2 df-mpt 5158 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
32rneqi 5846 . . 3 ran (𝑥𝐴𝐵) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
4 df-rex 3070 . . . 4 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥𝐴𝑦 = 𝐵))
54abbii 2808 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
61, 3, 53eqtr4i 2776 . 2 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
7 abrexexd.1 . . 3 (𝜑𝐴 ∈ V)
8 funmpt 6472 . . . 4 Fun (𝑥𝐴𝐵)
9 eqid 2738 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
109dmmpt 6143 . . . . 5 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
11 abrexexd.0 . . . . . 6 𝑥𝐴
1211rabexgfGS 30846 . . . . 5 (𝐴 ∈ V → {𝑥𝐴𝐵 ∈ V} ∈ V)
1310, 12eqeltrid 2843 . . . 4 (𝐴 ∈ V → dom (𝑥𝐴𝐵) ∈ V)
14 funex 7095 . . . 4 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
158, 13, 14sylancr 587 . . 3 (𝐴 ∈ V → (𝑥𝐴𝐵) ∈ V)
16 rnexg 7751 . . 3 ((𝑥𝐴𝐵) ∈ V → ran (𝑥𝐴𝐵) ∈ V)
177, 15, 163syl 18 . 2 (𝜑 → ran (𝑥𝐴𝐵) ∈ V)
186, 17eqeltrrid 2844 1 (𝜑 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wnfc 2887  wrex 3065  {crab 3068  Vcvv 3432  {copab 5136  cmpt 5157  dom cdm 5589  ran crn 5590  Fun wfun 6427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441
This theorem is referenced by:  esumc  32019
  Copyright terms: Public domain W3C validator