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Theorem dmss 4741
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 3091 . . . 4 (𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
21eximdv 1852 . . 3 (𝐴𝐵 → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
3 vex 2689 . . . 4 𝑥 ∈ V
43eldm2 4740 . . 3 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
53eldm2 4740 . . 3 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵)
62, 4, 53imtr4g 204 . 2 (𝐴𝐵 → (𝑥 ∈ dom 𝐴𝑥 ∈ dom 𝐵))
76ssrdv 3103 1 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1468  wcel 1480  wss 3071  cop 3530  dom cdm 4542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3933  df-dm 4552
This theorem is referenced by:  dmeq  4742  dmv  4758  rnss  4772  dmiin  4788  dmxpss2  4974  ssxpbm  4977  ssxp1  4978  cocnvres  5066  relrelss  5068  funssxp  5295  fvun1  5490  fndmdif  5528  fneqeql2  5532  tposss  6146  smores  6192  smores2  6194  tfrlemibfn  6228  tfrlemiubacc  6230  tfr1onlembfn  6244  tfr1onlemubacc  6246  tfr1onlemres  6249  tfrcllembfn  6257  tfrcllemubacc  6259  tfrcllemres  6262  frecuzrdgtcl  10209  frecuzrdgdomlem  10214  ennnfonelemex  11950  strleund  12073  strleun  12074  dvbssntrcntop  12848
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