Theorem dmss 4980

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 3242	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2	1	eximdv 1933	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		vex 2824	. . . 4 ⊢ 𝑥 ∈ V
4	3	eldm2 4979	. . 3 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
5	3	eldm2 4979	. . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	2, 4, 5	3imtr4g 205	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ dom 𝐴 → 𝑥 ∈ dom 𝐵))
7	6	ssrdv 3254	1 ⊢ (𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵)

This proof depends on syntax axioms:   → wi 4  ∃wex 1545   ∈ wcel 2209   ⊆ wss 3220  ⟨cop 3712  dom cdm 4774

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-dm 4784

This theorem is used by:  dmeq  4981  dmv  4997  rnss  5012  dmiin  5028  dmxpss2  5220  ssxpbm  5223  ssxp1  5224  cocnvres  5312  relrelss  5314  funssxp  5557  fvun1  5769  fndmdif  5814  fneqeql2  5818  funsssuppss  6498  tposss  6517  smores  6563  smores2  6565  tfrlemibfn  6599  tfrlemiubacc  6601  tfr1onlembfn  6615  tfr1onlemubacc  6617  tfr1onlemres  6620  tfrcllembfn  6628  tfrcllemubacc  6630  tfrcllemres  6633  frecuzrdgtcl  10851  frecuzrdgdomlem  10856  hashdmprop2dom  11298  ennnfonelemex  13307  strleund  13459  strleun  13460  imasaddfnlemg  13637  dvbssntrcntop  15787  subgreldmiedg  16522