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Theorem iinexd 39632
 Description: The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
iinexd.1 (𝜑𝐴 ≠ ∅)
iinexd.2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
Assertion
Ref Expression
iinexd (𝜑 𝑥𝐴 𝐵 ∈ V)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem iinexd
StepHypRef Expression
1 iinexd.1 . 2 (𝜑𝐴 ≠ ∅)
2 iinexd.2 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
3 iinexg 4854 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
41, 2, 3syl2anc 694 1 (𝜑 𝑥𝐴 𝐵 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  Vcvv 3231  ∅c0 3948  ∩ ciin 4553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-int 4508  df-iin 4555 This theorem is referenced by:  smfsuplem1  41338  smfinflem  41344  smflimsuplem1  41347  smflimsuplem2  41348  smflimsuplem3  41349  smflimsuplem4  41350  smflimsuplem5  41351  smflimsuplem7  41353
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