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Theorem abrexexd 30202
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 5825 . . 3 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
2 df-mpt 5144 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
32rneqi 5806 . . 3 ran (𝑥𝐴𝐵) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
4 df-rex 3149 . . . 4 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥𝐴𝑦 = 𝐵))
54abbii 2891 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
61, 3, 53eqtr4i 2859 . 2 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
7 abrexexd.1 . . 3 (𝜑𝐴 ∈ V)
8 funmpt 6392 . . . 4 Fun (𝑥𝐴𝐵)
9 eqid 2826 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
109dmmpt 6093 . . . . 5 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
11 abrexexd.0 . . . . . 6 𝑥𝐴
1211rabexgfGS 30195 . . . . 5 (𝐴 ∈ V → {𝑥𝐴𝐵 ∈ V} ∈ V)
1310, 12eqeltrid 2922 . . . 4 (𝐴 ∈ V → dom (𝑥𝐴𝐵) ∈ V)
14 funex 6979 . . . 4 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
158, 13, 14sylancr 587 . . 3 (𝐴 ∈ V → (𝑥𝐴𝐵) ∈ V)
16 rnexg 7607 . . 3 ((𝑥𝐴𝐵) ∈ V → ran (𝑥𝐴𝐵) ∈ V)
177, 15, 163syl 18 . 2 (𝜑 → ran (𝑥𝐴𝐵) ∈ V)
186, 17eqeltrrid 2923 1 (𝜑 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wex 1773  wcel 2107  {cab 2804  wnfc 2966  wrex 3144  {crab 3147  Vcvv 3500  {copab 5125  cmpt 5143  dom cdm 5554  ran crn 5555  Fun wfun 6348
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362
This theorem is referenced by:  esumc  31215
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