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Theorem iinexg 5230
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 4943 . . 3 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
21adantl 485 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3 elisset 3491 . . . . . . . . 9 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
43rgenw 3145 . . . . . . . 8 𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
5 r19.2z 4423 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)) → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
64, 5mpan2 690 . . . . . . 7 (𝐴 ≠ ∅ → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
7 r19.35 3332 . . . . . . 7 (∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵) ↔ (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
86, 7sylib 221 . . . . . 6 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
98imp 410 . . . . 5 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑥𝐴𝑦 𝑦 = 𝐵)
10 rexcom4 3243 . . . . 5 (∃𝑥𝐴𝑦 𝑦 = 𝐵 ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
119, 10sylib 221 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑦𝑥𝐴 𝑦 = 𝐵)
12 abn0 4319 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≠ ∅ ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
1311, 12sylibr 237 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≠ ∅)
14 intex 5226 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≠ ∅ ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
1513, 14sylib 221 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
162, 15eqeltrd 2916 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wne 3014  wral 3133  wrex 3134  Vcvv 3480  c0 4276   cint 4862   ciin 4906
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-in 3926  df-ss 3936  df-nul 4277  df-int 4863  df-iin 4908
This theorem is referenced by:  fclsval  22616  taylfval  24957  iinexd  41691  smflimlem1  43330  smfliminflem  43387
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