Theorem iinconstm 3657

Description: Indexed intersection of a constant class, i.e. where B does not depend on x. (Contributed by Jim Kingdon, 19-Dec-2018.)

Assertion

Ref	Expression
iinconstm	⊢ (∃y y ∈ A → ∩ x ∈ A B = B)

Distinct variable groups:   x,A   x,B   y,A

Allowed substitution hint:   B(y)

Proof of Theorem iinconstm

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.3rmv 3306	. . 3 ⊢ (∃y y ∈ A → (z ∈ B ↔ ∀x ∈ A z ∈ B))
2		vex 2554	. . . 4 ⊢ z ∈ V
3		eliin 3653	. . . 4 ⊢ (z ∈ V → (z ∈ ∩ x ∈ A B ↔ ∀x ∈ A z ∈ B))
4	2, 3	ax-mp 7	. . 3 ⊢ (z ∈ ∩ x ∈ A B ↔ ∀x ∈ A z ∈ B)
5	1, 4	syl6rbbr 188	. 2 ⊢ (∃y y ∈ A → (z ∈ ∩ x ∈ A B ↔ z ∈ B))
6	5	eqrdv 2035	1 ⊢ (∃y y ∈ A → ∩ x ∈ A B = B)

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  Vcvv 2551  ∩ ciin 3649

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-iin 3651

This theorem is referenced by:  iin0imm  3912