Theorem iinexd 45563

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 5289	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)
4	1, 2, 3	syl2anc 585	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  Vcvv 3429  ∅c0 4273  ∩ ciin 4934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-in 3896  df-ss 3906  df-nul 4274  df-int 4890  df-iin 4936

This theorem is referenced by:  smfsuplem1  47239  smfinflem  47245  smflimsuplem1  47248  smflimsuplem2  47249  smflimsuplem3  47250  smflimsuplem4  47251  smflimsuplem5  47252  smflimsuplem7  47254