Theorem iinconstm 3666

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 3312	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝑧 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
2		vex 2560	. . . 4 ⊢ 𝑧 ∈ V
3		eliin 3662	. . . 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 2038	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  Vcvv 2557  ∩ ciin 3658

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-iin 3660

This theorem is referenced by:  iin0imm  3921