Theorem iinconstm 3973

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 2802	. . . 4 ⊢ 𝑧 ∈ V
2		eliin 3969	. . . 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 3582	. . 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 2227	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  Vcvv 2799  ∩ ciin 3965

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-iin 3967

This theorem is referenced by:  iin0imm  4251