Theorem iinconstm 3979

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		vex 2805	. . . 4 ⊢ 𝑧 ∈ V
2		eliin 3975	. . . 4 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
3	1, 2	ax-mp 5	. . 3 ⊢ (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
4		r19.3rmv 3585	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝑧 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
5	3, 4	bitr4id 199	. 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ 𝐵))
6	5	eqrdv 2229	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  Vcvv 2802  ∩ ciin 3971

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-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-iin 3973

This theorem is referenced by:  iin0imm  4258