Theorem iinexd 45675

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 5303	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)
4	1, 2, 3	syl2anc 593	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  Vcvv 3453  ∅c0 4285  ∩ ciin 4949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-in 3911  df-ss 3921  df-nul 4286  df-int 4905  df-iin 4951

This theorem is referenced by:  smfsuplem1  47349  smfinflem  47355  smflimsuplem1  47358  smflimsuplem2  47359  smflimsuplem3  47360  smflimsuplem4  47361  smflimsuplem5  47362  smflimsuplem7  47364