Theorem iinexd 45831

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 5320	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)
4	1, 2, 3	syl2anc 595	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  Vcvv 3455  ∅c0 4287  ∩ ciin 4958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-in 3913  df-ss 3923  df-nul 4288  df-int 4914  df-iin 4960

This theorem is referenced by:  smfsuplem1  47505  smfinflem  47511  smflimsuplem1  47514  smflimsuplem2  47515  smflimsuplem3  47516  smflimsuplem4  47517  smflimsuplem5  47518  smflimsuplem7  47520