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Theorem dmss 4593
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 3004 . . . 4 (𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
21eximdv 1803 . . 3 (𝐴𝐵 → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
3 vex 2615 . . . 4 𝑥 ∈ V
43eldm2 4592 . . 3 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
53eldm2 4592 . . 3 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵)
62, 4, 53imtr4g 203 . 2 (𝐴𝐵 → (𝑥 ∈ dom 𝐴𝑥 ∈ dom 𝐵))
76ssrdv 3016 1 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1422  wcel 1434  wss 2984  cop 3425  dom cdm 4401
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-dm 4411
This theorem is referenced by:  dmeq  4594  dmv  4610  rnss  4623  dmiin  4639  dmxpss2  4817  ssxpbm  4820  ssxp1  4821  cocnvres  4909  relrelss  4911  funssxp  5129  fvun1  5315  fndmdif  5349  fneqeql2  5353  tposss  5943  smores  5989  smores2  5991  tfrlemibfn  6025  tfrlemiubacc  6027  tfr1onlembfn  6041  tfr1onlemubacc  6043  tfr1onlemres  6046  tfrcllembfn  6054  tfrcllemubacc  6056  tfrcllemres  6059  frecuzrdgtcl  9708  frecuzrdgdomlem  9713
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