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Theorem dminss 6175
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 2179 . . . . . . 7 ((𝑥𝐴𝑥𝑅𝑦) → ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
21ancoms 458 . . . . . 6 ((𝑥𝑅𝑦𝑥𝐴) → ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
3 vex 3482 . . . . . . 7 𝑦 ∈ V
43elima2 6086 . . . . . 6 (𝑦 ∈ (𝑅𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
52, 4sylibr 234 . . . . 5 ((𝑥𝑅𝑦𝑥𝐴) → 𝑦 ∈ (𝑅𝐴))
6 simpl 482 . . . . . 6 ((𝑥𝑅𝑦𝑥𝐴) → 𝑥𝑅𝑦)
7 vex 3482 . . . . . . 7 𝑥 ∈ V
83, 7brcnv 5896 . . . . . 6 (𝑦𝑅𝑥𝑥𝑅𝑦)
96, 8sylibr 234 . . . . 5 ((𝑥𝑅𝑦𝑥𝐴) → 𝑦𝑅𝑥)
105, 9jca 511 . . . 4 ((𝑥𝑅𝑦𝑥𝐴) → (𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
1110eximi 1832 . . 3 (∃𝑦(𝑥𝑅𝑦𝑥𝐴) → ∃𝑦(𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
127eldm 5914 . . . . 5 (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
1312anbi1i 624 . . . 4 ((𝑥 ∈ dom 𝑅𝑥𝐴) ↔ (∃𝑦 𝑥𝑅𝑦𝑥𝐴))
14 elin 3979 . . . 4 (𝑥 ∈ (dom 𝑅𝐴) ↔ (𝑥 ∈ dom 𝑅𝑥𝐴))
15 19.41v 1947 . . . 4 (∃𝑦(𝑥𝑅𝑦𝑥𝐴) ↔ (∃𝑦 𝑥𝑅𝑦𝑥𝐴))
1613, 14, 153bitr4i 303 . . 3 (𝑥 ∈ (dom 𝑅𝐴) ↔ ∃𝑦(𝑥𝑅𝑦𝑥𝐴))
177elima2 6086 . . 3 (𝑥 ∈ (𝑅 “ (𝑅𝐴)) ↔ ∃𝑦(𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
1811, 16, 173imtr4i 292 . 2 (𝑥 ∈ (dom 𝑅𝐴) → 𝑥 ∈ (𝑅 “ (𝑅𝐴)))
1918ssriv 3999 1 (dom 𝑅𝐴) ⊆ (𝑅 “ (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wa 395  wex 1776  wcel 2106  cin 3962  wss 3963   class class class wbr 5148  ccnv 5688  dom cdm 5689  cima 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by:  lmhmlsp  21066  cnclsi  23296  kgencn3  23582  kqsat  23755  kqcldsat  23757  cfilucfil  24588  elrspunidl  33436
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