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Theorem iinexgm 4172
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 3934 . . 3 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
21adantl 277 . 2 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3 elisset 2766 . . . . . . . . . 10 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
43rgenw 2545 . . . . . . . . 9 𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)
5 r19.2m 3524 . . . . . . . . 9 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵)) → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
64, 5mpan2 425 . . . . . . . 8 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵))
7 r19.35-1 2640 . . . . . . . 8 (∃𝑥𝐴 (𝐵𝐶 → ∃𝑦 𝑦 = 𝐵) → (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
86, 7syl 14 . . . . . . 7 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴𝑦 𝑦 = 𝐵))
98imp 124 . . . . . 6 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑥𝐴𝑦 𝑦 = 𝐵)
10 rexcom4 2775 . . . . . 6 (∃𝑥𝐴𝑦 𝑦 = 𝐵 ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
119, 10sylib 122 . . . . 5 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑦𝑥𝐴 𝑦 = 𝐵)
12 abid 2177 . . . . . 6 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑥𝐴 𝑦 = 𝐵)
1312exbii 1616 . . . . 5 (∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑦𝑥𝐴 𝑦 = 𝐵)
1411, 13sylibr 134 . . . 4 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
15 nfv 1539 . . . . 5 𝑧 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
16 nfsab1 2179 . . . . 5 𝑦 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
17 eleq1w 2250 . . . . 5 (𝑦 = 𝑧 → (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}))
1815, 16, 17cbvex 1767 . . . 4 (∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑧 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
1914, 18sylib 122 . . 3 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → ∃𝑧 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
20 inteximm 4167 . . 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 2266 1 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wex 1503  wcel 2160  {cab 2175  wral 2468  wrex 2469  Vcvv 2752   cint 3859   ciin 3902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-int 3860  df-iin 3904
This theorem is referenced by: (None)
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