Theorem dmss 4477

Description: Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)

Assertion

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

Proof of Theorem dmss

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

Step	Hyp	Ref	Expression
1		ssel 2933	. . . 4 ⊢ (A ⊆ B → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B))
2	1	eximdv 1757	. . 3 ⊢ (A ⊆ B → (∃y⟨x, y⟩ ∈ A → ∃y⟨x, y⟩ ∈ B))
3		vex 2554	. . . 4 ⊢ x ∈ V
4	3	eldm2 4476	. . 3 ⊢ (x ∈ dom A ↔ ∃y⟨x, y⟩ ∈ A)
5	3	eldm2 4476	. . 3 ⊢ (x ∈ dom B ↔ ∃y⟨x, y⟩ ∈ B)
6	2, 4, 5	3imtr4g 194	. 2 ⊢ (A ⊆ B → (x ∈ dom A → x ∈ dom B))
7	6	ssrdv 2945	1 ⊢ (A ⊆ B → dom A ⊆ dom B)

Syntax hints:   → wi 4  ∃wex 1378   ∈ wcel 1390   ⊆ 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:  dmeq  4478  dmv  4494  rnss  4507  dmiin  4523  ssxpbm  4699  ssxp1  4700  relrelss  4787  funssxp  5003  fvun1  5182  fndmdif  5215  fneqeql2  5219  tposss  5802  smores  5848  smores2  5850  tfrlemibfn  5883  tfrlemiubacc  5885  frecuzrdgfn  8879