Theorem iinexd 46241

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 5308	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)
4	1, 2, 3	syl2anc 596	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450  ∅c0 4278  ∩ ciin 4951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-nul 4279  df-int 4907  df-iin 4953

This theorem is used by:  smfsuplem1  47914  smfinflem  47920  smflimsuplem1  47923  smflimsuplem2  47924  smflimsuplem3  47925  smflimsuplem4  47926  smflimsuplem5  47927  smflimsuplem7  47929