Theorem dmss 4676

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 3041	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2	1	eximdv 1819	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		vex 2644	. . . 4 ⊢ 𝑥 ∈ V
4	3	eldm2 4675	. . 3 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
5	3	eldm2 4675	. . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	2, 4, 5	3imtr4g 204	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ dom 𝐴 → 𝑥 ∈ dom 𝐵))
7	6	ssrdv 3053	1 ⊢ (𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵)

Syntax hints:   → wi 4  ∃wex 1436   ∈ wcel 1448   ⊆ wss 3021  ⟨cop 3477  dom cdm 4477

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-dm 4487

This theorem is referenced by:  dmeq  4677  dmv  4693  rnss  4707  dmiin  4723  dmxpss2  4907  ssxpbm  4910  ssxp1  4911  cocnvres  4999  relrelss  5001  funssxp  5228  fvun1  5419  fndmdif  5457  fneqeql2  5461  tposss  6073  smores  6119  smores2  6121  tfrlemibfn  6155  tfrlemiubacc  6157  tfr1onlembfn  6171  tfr1onlemubacc  6173  tfr1onlemres  6176  tfrcllembfn  6184  tfrcllemubacc  6186  tfrcllemres  6189  frecuzrdgtcl  10026  frecuzrdgdomlem  10031  ennnfonelemex  11719  strleund  11829  strleun  11830  dvbssntrcntop  12526