Theorem dmin 4486

Description: The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Assertion

Ref	Expression
dmin	⊢ dom (A ∩ B) ⊆ (dom A ∩ dom B)

Proof of Theorem dmin

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.40 1519	. . 3 ⊢ (∃y(⟨x, y⟩ ∈ A ∧ ⟨x, y⟩ ∈ B) → (∃y⟨x, y⟩ ∈ A ∧ ∃y⟨x, y⟩ ∈ B))
2		vex 2554	. . . . 5 ⊢ x ∈ V
3	2	eldm2 4476	. . . 4 ⊢ (x ∈ dom (A ∩ B) ↔ ∃y⟨x, y⟩ ∈ (A ∩ B))
4		elin 3120	. . . . 5 ⊢ (⟨x, y⟩ ∈ (A ∩ B) ↔ (⟨x, y⟩ ∈ A ∧ ⟨x, y⟩ ∈ B))
5	4	exbii 1493	. . . 4 ⊢ (∃y⟨x, y⟩ ∈ (A ∩ B) ↔ ∃y(⟨x, y⟩ ∈ A ∧ ⟨x, y⟩ ∈ B))
6	3, 5	bitri 173	. . 3 ⊢ (x ∈ dom (A ∩ B) ↔ ∃y(⟨x, y⟩ ∈ A ∧ ⟨x, y⟩ ∈ B))
7		elin 3120	. . . 4 ⊢ (x ∈ (dom A ∩ dom B) ↔ (x ∈ dom A ∧ x ∈ dom B))
8	2	eldm2 4476	. . . . 5 ⊢ (x ∈ dom A ↔ ∃y⟨x, y⟩ ∈ A)
9	2	eldm2 4476	. . . . 5 ⊢ (x ∈ dom B ↔ ∃y⟨x, y⟩ ∈ B)
10	8, 9	anbi12i 433	. . . 4 ⊢ ((x ∈ dom A ∧ x ∈ dom B) ↔ (∃y⟨x, y⟩ ∈ A ∧ ∃y⟨x, y⟩ ∈ B))
11	7, 10	bitri 173	. . 3 ⊢ (x ∈ (dom A ∩ dom B) ↔ (∃y⟨x, y⟩ ∈ A ∧ ∃y⟨x, y⟩ ∈ B))
12	1, 6, 11	3imtr4i 190	. 2 ⊢ (x ∈ dom (A ∩ B) → x ∈ (dom A ∩ dom B))
13	12	ssriv 2943	1 ⊢ dom (A ∩ B) ⊆ (dom A ∩ dom B)

Syntax hints:   ∧ wa 97  ∃wex 1378   ∈ wcel 1390   ∩ cin 2910   ⊆ wss 2911  ⟨cop 3370  dom cdm 4288

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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-dm 4298

This theorem is referenced by:  rnin  4676