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Theorem iinexg 5235
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 4948 . . 3 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
21adantl 482 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3 elisset 3503 . . . . . . . . 9 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
43rgenw 3147 . . . . . . . 8 𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
5 r19.2z 4436 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)) → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
64, 5mpan2 687 . . . . . . 7 (𝐴 ≠ ∅ → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
7 r19.35 3338 . . . . . . 7 (∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵) ↔ (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
86, 7sylib 219 . . . . . 6 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
98imp 407 . . . . 5 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑥𝐴𝑦 𝑦 = 𝐵)
10 rexcom4 3246 . . . . 5 (∃𝑥𝐴𝑦 𝑦 = 𝐵 ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
119, 10sylib 219 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑦𝑥𝐴 𝑦 = 𝐵)
12 abn0 4333 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≠ ∅ ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
1311, 12sylibr 235 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≠ ∅)
14 intex 5231 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≠ ∅ ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
1513, 14sylib 219 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
162, 15eqeltrd 2910 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wne 3013  wral 3135  wrex 3136  Vcvv 3492  c0 4288   cint 4867   ciin 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-int 4868  df-iin 4913
This theorem is referenced by:  fclsval  22544  taylfval  24874  iinexd  41276  smflimlem1  42924  smfliminflem  42981
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