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Theorem dfint3 32390
Description: Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)
Assertion
Ref Expression
dfint3 𝐴 = (V ∖ ((V ∖ E ) “ 𝐴))

Proof of Theorem dfint3
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfint2 4611 . 2 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
2 ralnex 3140 . . . 4 (∀𝑦𝐴 ¬ 𝑦(V ∖ E )𝑥 ↔ ¬ ∃𝑦𝐴 𝑦(V ∖ E )𝑥)
3 vex 3352 . . . . . . . . 9 𝑦 ∈ V
4 vex 3352 . . . . . . . . 9 𝑥 ∈ V
53, 4brcnv 5443 . . . . . . . 8 (𝑦(V ∖ E )𝑥𝑥(V ∖ E )𝑦)
6 brv 5068 . . . . . . . . 9 𝑥V𝑦
7 brdif 4837 . . . . . . . . 9 (𝑥(V ∖ E )𝑦 ↔ (𝑥V𝑦 ∧ ¬ 𝑥 E 𝑦))
86, 7mpbiran 680 . . . . . . . 8 (𝑥(V ∖ E )𝑦 ↔ ¬ 𝑥 E 𝑦)
95, 8bitr2i 265 . . . . . . 7 𝑥 E 𝑦𝑦(V ∖ E )𝑥)
109con1bii 345 . . . . . 6 𝑦(V ∖ E )𝑥𝑥 E 𝑦)
11 epel 5165 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
1210, 11bitr2i 265 . . . . 5 (𝑥𝑦 ↔ ¬ 𝑦(V ∖ E )𝑥)
1312ralbii 3128 . . . 4 (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦(V ∖ E )𝑥)
14 eldif 3731 . . . . . 6 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ E ) “ 𝐴)))
154, 14mpbiran 680 . . . . 5 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ ¬ 𝑥 ∈ ((V ∖ E ) “ 𝐴))
164elima 5612 . . . . 5 (𝑥 ∈ ((V ∖ E ) “ 𝐴) ↔ ∃𝑦𝐴 𝑦(V ∖ E )𝑥)
1715, 16xchbinx 323 . . . 4 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ ¬ ∃𝑦𝐴 𝑦(V ∖ E )𝑥)
182, 13, 173bitr4ri 293 . . 3 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ ∀𝑦𝐴 𝑥𝑦)
1918abbi2i 2886 . 2 (V ∖ ((V ∖ E ) “ 𝐴)) = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
201, 19eqtr4i 2795 1 𝐴 = (V ∖ ((V ∖ E ) “ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1630  wcel 2144  {cab 2756  wral 3060  wrex 3061  Vcvv 3349  cdif 3718   cint 4609   class class class wbr 4784   E cep 5161  ccnv 5248  cima 5252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-int 4610  df-br 4785  df-opab 4845  df-eprel 5162  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262
This theorem is referenced by: (None)
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