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Theorem dmss 4605
Description: Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
Assertion
Ref Expression
dmss (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)

Proof of Theorem dmss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3008 . . . 4 (𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
21eximdv 1805 . . 3 (𝐴𝐵 → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
3 vex 2618 . . . 4 𝑥 ∈ V
43eldm2 4604 . . 3 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
53eldm2 4604 . . 3 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵)
62, 4, 53imtr4g 203 . 2 (𝐴𝐵 → (𝑥 ∈ dom 𝐴𝑥 ∈ dom 𝐵))
76ssrdv 3020 1 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1424  wcel 1436  wss 2988  cop 3434  dom cdm 4413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-dm 4423
This theorem is referenced by:  dmeq  4606  dmv  4622  rnss  4635  dmiin  4651  dmxpss2  4831  ssxpbm  4834  ssxp1  4835  cocnvres  4923  relrelss  4925  funssxp  5146  fvun1  5335  fndmdif  5369  fneqeql2  5373  tposss  5967  smores  6013  smores2  6015  tfrlemibfn  6049  tfrlemiubacc  6051  tfr1onlembfn  6065  tfr1onlemubacc  6067  tfr1onlemres  6070  tfrcllembfn  6078  tfrcllemubacc  6080  tfrcllemres  6083  frecuzrdgtcl  9750  frecuzrdgdomlem  9755
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