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Theorem iinexgm 3905
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 3687 . . 3 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
21adantl 262 . 2 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3 elisset 2565 . . . . . . . . . 10 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
43rgenw 2373 . . . . . . . . 9 𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
5 r19.2m 3306 . . . . . . . . 9 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)) → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
64, 5mpan2 401 . . . . . . . 8 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
7 r19.35-1 2457 . . . . . . . 8 (∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵) → (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
86, 7syl 14 . . . . . . 7 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
98imp 115 . . . . . 6 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑥𝐴𝑦 𝑦 = 𝐵)
10 rexcom4 2574 . . . . . 6 (∃𝑥𝐴𝑦 𝑦 = 𝐵 ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
119, 10sylib 127 . . . . 5 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑦𝑥𝐴 𝑦 = 𝐵)
12 abid 2028 . . . . . 6 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑥𝐴 𝑦 = 𝐵)
1312exbii 1496 . . . . 5 (∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
1411, 13sylibr 137 . . . 4 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
15 nfv 1421 . . . . 5 𝑧 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
16 nfsab1 2030 . . . . 5 𝑦 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
17 eleq1 2100 . . . . 5 (𝑦 = 𝑧 → (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}))
1815, 16, 17cbvex 1639 . . . 4 (∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑧 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
1914, 18sylib 127 . . 3 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑧 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
20 inteximm 3900 . . 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 2114 1 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wex 1381  wcel 1393  {cab 2026  wral 2303  wrex 2304  Vcvv 2554   cint 3612   ciin 3655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-in 2921  df-ss 2928  df-int 3613  df-iin 3657
This theorem is referenced by: (None)
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