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Theorem iinexg 4961
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 4693 . . 3 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
21adantl 473 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3 elisset 3343 . . . . . . . . 9 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
43rgenw 3050 . . . . . . . 8 𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
5 r19.2z 4192 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)) → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
64, 5mpan2 709 . . . . . . 7 (𝐴 ≠ ∅ → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
7 r19.35 3210 . . . . . . 7 (∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵) ↔ (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
86, 7sylib 208 . . . . . 6 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
98imp 444 . . . . 5 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑥𝐴𝑦 𝑦 = 𝐵)
10 rexcom4 3353 . . . . 5 (∃𝑥𝐴𝑦 𝑦 = 𝐵 ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
119, 10sylib 208 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑦𝑥𝐴 𝑦 = 𝐵)
12 abn0 4085 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≠ ∅ ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
1311, 12sylibr 224 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≠ ∅)
14 intex 4957 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≠ ∅ ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
1513, 14sylib 208 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
162, 15eqeltrd 2827 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wex 1841  wcel 2127  {cab 2734  wne 2920  wral 3038  wrex 3039  Vcvv 3328  c0 4046   cint 4615   ciin 4661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-v 3330  df-dif 3706  df-in 3710  df-ss 3717  df-nul 4047  df-int 4616  df-iin 4663
This theorem is referenced by:  fclsval  21984  taylfval  24283  iinexd  39786  smflimlem1  41454  smfliminflem  41511
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