Theorem dmss 4885

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 3191	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2	1	eximdv 1904	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		vex 2776	. . . 4 ⊢ 𝑥 ∈ V
4	3	eldm2 4884	. . 3 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
5	3	eldm2 4884	. . 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 3203	1 ⊢ (𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵)

Syntax hints:   → wi 4  ∃wex 1516   ∈ wcel 2177   ⊆ wss 3170  ⟨cop 3640  dom cdm 4682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3643  df-pr 3644  df-op 3646  df-br 4051  df-dm 4692

This theorem is referenced by:  dmeq  4886  dmv  4902  rnss  4916  dmiin  4932  dmxpss2  5123  ssxpbm  5126  ssxp1  5127  cocnvres  5215  relrelss  5217  funssxp  5454  fvun1  5657  fndmdif  5697  fneqeql2  5701  tposss  6344  smores  6390  smores2  6392  tfrlemibfn  6426  tfrlemiubacc  6428  tfr1onlembfn  6442  tfr1onlemubacc  6444  tfr1onlemres  6447  tfrcllembfn  6455  tfrcllemubacc  6457  tfrcllemres  6460  frecuzrdgtcl  10574  frecuzrdgdomlem  10579  hashdmprop2dom  11006  ennnfonelemex  12855  strleund  13005  strleun  13006  imasaddfnlemg  13216  dvbssntrcntop  15226