Theorem iinconstm 3640

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 3290	. . 3 ⊢ (∃y y ∈ A → (z ∈ B ↔ ∀x ∈ A z ∈ B))
2		vex 2538	. . . 4 ⊢ z ∈ V
3		eliin 3636	. . . 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 2020	1 ⊢ (∃y y ∈ A → ∩ x ∈ A B = B)

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∀wral 2284  Vcvv 2535  ∩ ciin 3632

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-iin 3634

This theorem is referenced by:  iin0imm  3895