Theorem dmss 2530

Description: Subset theorem for domain.

Assertion

Ref	Expression
dmss	⊢ (A ⊆ B → dom A ⊆ dom B)

Proof of Theorem dmss

Step	Hyp	Ref	Expression
1		ssel 1502	. . . 4 ⊢ (A ⊆ B → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B))
2	1	19.22dv 947	. . 3 ⊢ (A ⊆ B → (∃y⟨x, y⟩ ∈ A → ∃y⟨x, y⟩ ∈ B))
3		visset 1350	. . . 4 ⊢ x ∈ V
4	3	eldm2 2528	. . 3 ⊢ (x ∈ dom A ↔ ∃y⟨x, y⟩ ∈ A)
5	3	eldm2 2528	. . 3 ⊢ (x ∈ dom B ↔ ∃y⟨x, y⟩ ∈ B)
6	2, 4, 5	3imtr4g 426	. 2 ⊢ (A ⊆ B → (x ∈ dom A → x ∈ dom B))
7	6	ssrdv 1509	1 ⊢ (A ⊆ B → dom A ⊆ dom B)

Syntax hints:   → wi 2  ∃wex 678   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810  dom cdm 2410

This theorem is referenced by:  dmeq 2531  dmv 2546  rnss 2558  tfrlem8 2956  tfrlem13 2961  ecopoprdm 3245  dmen 3310

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-dm 2428

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