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Theorem iinexgm 3958
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 3740 . . 3 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
21adantl 271 . 2 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3 elisset 2626 . . . . . . . . . 10 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
43rgenw 2426 . . . . . . . . 9 𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
5 r19.2m 3353 . . . . . . . . 9 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)) → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
64, 5mpan2 416 . . . . . . . 8 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
7 r19.35-1 2512 . . . . . . . 8 (∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵) → (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
86, 7syl 14 . . . . . . 7 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
98imp 122 . . . . . 6 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑥𝐴𝑦 𝑦 = 𝐵)
10 rexcom4 2635 . . . . . 6 (∃𝑥𝐴𝑦 𝑦 = 𝐵 ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
119, 10sylib 120 . . . . 5 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑦𝑥𝐴 𝑦 = 𝐵)
12 abid 2073 . . . . . 6 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑥𝐴 𝑦 = 𝐵)
1312exbii 1539 . . . . 5 (∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
1411, 13sylibr 132 . . . 4 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
15 nfv 1464 . . . . 5 𝑧 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
16 nfsab1 2075 . . . . 5 𝑦 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
17 eleq1 2147 . . . . 5 (𝑦 = 𝑧 → (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}))
1815, 16, 17cbvex 1683 . . . 4 (∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑧 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
1914, 18sylib 120 . . 3 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑧 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
20 inteximm 3953 . . 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 2161 1 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wex 1424  wcel 1436  {cab 2071  wral 2355  wrex 2356  Vcvv 2614   cint 3665   ciin 3708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-in 2992  df-ss 2999  df-int 3666  df-iin 3710
This theorem is referenced by: (None)
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