ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iinexgm GIF version

Theorem iinexgm 4079
Description: The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by Jim Kingdon, 28-Aug-2018.)
Assertion
Ref Expression
iinexgm ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem iinexgm
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfiin2g 3846 . . 3 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
21adantl 275 . 2 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3 elisset 2700 . . . . . . . . . 10 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
43rgenw 2487 . . . . . . . . 9 𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
5 r19.2m 3449 . . . . . . . . 9 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)) → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
64, 5mpan2 421 . . . . . . . 8 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
7 r19.35-1 2581 . . . . . . . 8 (∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵) → (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
86, 7syl 14 . . . . . . 7 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
98imp 123 . . . . . 6 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑥𝐴𝑦 𝑦 = 𝐵)
10 rexcom4 2709 . . . . . 6 (∃𝑥𝐴𝑦 𝑦 = 𝐵 ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
119, 10sylib 121 . . . . 5 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑦𝑥𝐴 𝑦 = 𝐵)
12 abid 2127 . . . . . 6 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑥𝐴 𝑦 = 𝐵)
1312exbii 1584 . . . . 5 (∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
1411, 13sylibr 133 . . . 4 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
15 nfv 1508 . . . . 5 𝑧 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
16 nfsab1 2129 . . . . 5 𝑦 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
17 eleq1w 2200 . . . . 5 (𝑦 = 𝑧 → (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}))
1815, 16, 17cbvex 1729 . . . 4 (∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑧 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
1914, 18sylib 121 . . 3 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑧 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
20 inteximm 4074 . . 3 (∃𝑧 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
2119, 20syl 14 . 2 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
222, 21eqeltrd 2216 1 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wex 1468  wcel 1480  {cab 2125  wral 2416  wrex 2417  Vcvv 2686   cint 3771   ciin 3814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-int 3772  df-iin 3816
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator