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Theorem dminss 6173
Description: An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising". (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
dminss (dom 𝑅𝐴) ⊆ (𝑅 “ (𝑅𝐴))

Proof of Theorem dminss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.8a 2181 . . . . . . 7 ((𝑥𝐴𝑥𝑅𝑦) → ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
21ancoms 458 . . . . . 6 ((𝑥𝑅𝑦𝑥𝐴) → ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
3 vex 3484 . . . . . . 7 𝑦 ∈ V
43elima2 6084 . . . . . 6 (𝑦 ∈ (𝑅𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
52, 4sylibr 234 . . . . 5 ((𝑥𝑅𝑦𝑥𝐴) → 𝑦 ∈ (𝑅𝐴))
6 simpl 482 . . . . . 6 ((𝑥𝑅𝑦𝑥𝐴) → 𝑥𝑅𝑦)
7 vex 3484 . . . . . . 7 𝑥 ∈ V
83, 7brcnv 5893 . . . . . 6 (𝑦𝑅𝑥𝑥𝑅𝑦)
96, 8sylibr 234 . . . . 5 ((𝑥𝑅𝑦𝑥𝐴) → 𝑦𝑅𝑥)
105, 9jca 511 . . . 4 ((𝑥𝑅𝑦𝑥𝐴) → (𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
1110eximi 1835 . . 3 (∃𝑦(𝑥𝑅𝑦𝑥𝐴) → ∃𝑦(𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
127eldm 5911 . . . . 5 (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
1312anbi1i 624 . . . 4 ((𝑥 ∈ dom 𝑅𝑥𝐴) ↔ (∃𝑦 𝑥𝑅𝑦𝑥𝐴))
14 elin 3967 . . . 4 (𝑥 ∈ (dom 𝑅𝐴) ↔ (𝑥 ∈ dom 𝑅𝑥𝐴))
15 19.41v 1949 . . . 4 (∃𝑦(𝑥𝑅𝑦𝑥𝐴) ↔ (∃𝑦 𝑥𝑅𝑦𝑥𝐴))
1613, 14, 153bitr4i 303 . . 3 (𝑥 ∈ (dom 𝑅𝐴) ↔ ∃𝑦(𝑥𝑅𝑦𝑥𝐴))
177elima2 6084 . . 3 (𝑥 ∈ (𝑅 “ (𝑅𝐴)) ↔ ∃𝑦(𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
1811, 16, 173imtr4i 292 . 2 (𝑥 ∈ (dom 𝑅𝐴) → 𝑥 ∈ (𝑅 “ (𝑅𝐴)))
1918ssriv 3987 1 (dom 𝑅𝐴) ⊆ (𝑅 “ (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wa 395  wex 1779  wcel 2108  cin 3950  wss 3951   class class class wbr 5143  ccnv 5684  dom cdm 5685  cima 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  lmhmlsp  21048  cnclsi  23280  kgencn3  23566  kqsat  23739  kqcldsat  23741  cfilucfil  24572  elrgspnsubrunlem2  33252  elrspunidl  33456
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