Theorem dfiin3 4992

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 4990	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
2		dfiun3.1	. . 3 ⊢ 𝐵 ∈ V
3	2	a1i 9	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ V)
4	1, 3	mprg 2589	1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   = wceq 1397   ∈ wcel 2202  Vcvv 2802  ∩ cint 3928  ∩ ciin 3971   ↦ cmpt 4150  ran crn 4726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-int 3929  df-iin 3973  df-br 4089  df-opab 4151  df-mpt 4152  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by: (None)