Theorem dfiin3 5984

Description: Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
dfiun3.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
dfiin3	⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵)

Proof of Theorem dfiin3

Step	Hyp	Ref	Expression
1		dfiin3g 5982	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
2		dfiun3.1	. . 3 ⊢ 𝐵 ∈ V
3	2	a1i 11	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ V)
4	1, 3	mprg 3065	1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∩ cint 4951  ∩ ciin 4997   ↦ cmpt 5231  ran crn 5690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-int 4952  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5697  df-dm 5699  df-rn 5700

This theorem is referenced by:  subdrgint  20821  fclscmpi  24053