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Theorem iinexg 5298
Description: The existence of a class intersection. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
iinexg ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem iinexg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfiin2g 4992 . . 3 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
21adantl 482 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3 elisset 2819 . . . . . . . . 9 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
43rgenw 3068 . . . . . . . 8 𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
5 r19.2z 4452 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)) → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
64, 5mpan2 689 . . . . . . 7 (𝐴 ≠ ∅ → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
7 r19.35 3111 . . . . . . 7 (∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵) ↔ (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
86, 7sylib 217 . . . . . 6 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
98imp 407 . . . . 5 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑥𝐴𝑦 𝑦 = 𝐵)
10 rexcom4 3271 . . . . 5 (∃𝑥𝐴𝑦 𝑦 = 𝐵 ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
119, 10sylib 217 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑦𝑥𝐴 𝑦 = 𝐵)
12 abn0 4340 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≠ ∅ ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
1311, 12sylibr 233 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≠ ∅)
14 intex 5294 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≠ ∅ ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
1513, 14sylib 217 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
162, 15eqeltrd 2838 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  Vcvv 3445  c0 4282   cint 4907   ciin 4955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-in 3917  df-ss 3927  df-nul 4283  df-int 4908  df-iin 4957
This theorem is referenced by:  fclsval  23359  taylfval  25718  iinexd  43333  smflimlem1  45002  smfliminflem  45061
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