Theorem iinexd 44499

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	⊢ (𝜑 → 𝐴 ≠ ∅)
iinexd.2	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)

Assertion

Ref	Expression
iinexd	⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem iinexd

Step	Hyp	Ref	Expression
1		iinexd.1	. 2 ⊢ (𝜑 → 𝐴 ≠ ∅)
2		iinexd.2	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
3		iinexg 5343	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)
4	1, 2, 3	syl2anc 583	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2099   ≠ wne 2937  ∀wral 3058  Vcvv 3471  ∅c0 4323  ∩ ciin 4997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4324  df-int 4950  df-iin 4999

This theorem is referenced by:  smfsuplem1  46199  smfinflem  46205  smflimsuplem1  46208  smflimsuplem2  46209  smflimsuplem3  46210  smflimsuplem4  46211  smflimsuplem5  46212  smflimsuplem7  46214