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Theorem abrexexd 29191
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 5330 . . 3 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
2 df-mpt 4675 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
32rneqi 5312 . . 3 ran (𝑥𝐴𝐵) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
4 df-rex 2913 . . . 4 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥𝐴𝑦 = 𝐵))
54abbii 2736 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
61, 3, 53eqtr4i 2653 . 2 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
7 abrexexd.1 . . 3 (𝜑𝐴 ∈ V)
8 funmpt 5884 . . . 4 Fun (𝑥𝐴𝐵)
9 eqid 2621 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
109dmmpt 5589 . . . . 5 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
11 abrexexd.0 . . . . . 6 𝑥𝐴
1211rabexgfGS 29185 . . . . 5 (𝐴 ∈ V → {𝑥𝐴𝐵 ∈ V} ∈ V)
1310, 12syl5eqel 2702 . . . 4 (𝐴 ∈ V → dom (𝑥𝐴𝐵) ∈ V)
14 funex 6436 . . . 4 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
158, 13, 14sylancr 694 . . 3 (𝐴 ∈ V → (𝑥𝐴𝐵) ∈ V)
16 rnexg 7045 . . 3 ((𝑥𝐴𝐵) ∈ V → ran (𝑥𝐴𝐵) ∈ V)
177, 15, 163syl 18 . 2 (𝜑 → ran (𝑥𝐴𝐵) ∈ V)
186, 17syl5eqelr 2703 1 (𝜑 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wnfc 2748  wrex 2908  {crab 2911  Vcvv 3186  {copab 4672  cmpt 4673  dom cdm 5074  ran crn 5075  Fun wfun 5841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855
This theorem is referenced by:  esumc  29891
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