ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dminss GIF version

Theorem dminss 5061
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 1601 . . . . . . 7 ((𝑥𝐴𝑥𝑅𝑦) → ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
21ancoms 268 . . . . . 6 ((𝑥𝑅𝑦𝑥𝐴) → ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
3 vex 2755 . . . . . . 7 𝑦 ∈ V
43elima2 4994 . . . . . 6 (𝑦 ∈ (𝑅𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
52, 4sylibr 134 . . . . 5 ((𝑥𝑅𝑦𝑥𝐴) → 𝑦 ∈ (𝑅𝐴))
6 simpl 109 . . . . . 6 ((𝑥𝑅𝑦𝑥𝐴) → 𝑥𝑅𝑦)
7 vex 2755 . . . . . . 7 𝑥 ∈ V
83, 7brcnv 4828 . . . . . 6 (𝑦𝑅𝑥𝑥𝑅𝑦)
96, 8sylibr 134 . . . . 5 ((𝑥𝑅𝑦𝑥𝐴) → 𝑦𝑅𝑥)
105, 9jca 306 . . . 4 ((𝑥𝑅𝑦𝑥𝐴) → (𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
1110eximi 1611 . . 3 (∃𝑦(𝑥𝑅𝑦𝑥𝐴) → ∃𝑦(𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
127eldm 4842 . . . . 5 (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
1312anbi1i 458 . . . 4 ((𝑥 ∈ dom 𝑅𝑥𝐴) ↔ (∃𝑦 𝑥𝑅𝑦𝑥𝐴))
14 elin 3333 . . . 4 (𝑥 ∈ (dom 𝑅𝐴) ↔ (𝑥 ∈ dom 𝑅𝑥𝐴))
15 19.41v 1914 . . . 4 (∃𝑦(𝑥𝑅𝑦𝑥𝐴) ↔ (∃𝑦 𝑥𝑅𝑦𝑥𝐴))
1613, 14, 153bitr4i 212 . . 3 (𝑥 ∈ (dom 𝑅𝐴) ↔ ∃𝑦(𝑥𝑅𝑦𝑥𝐴))
177elima2 4994 . . 3 (𝑥 ∈ (𝑅 “ (𝑅𝐴)) ↔ ∃𝑦(𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
1811, 16, 173imtr4i 201 . 2 (𝑥 ∈ (dom 𝑅𝐴) → 𝑥 ∈ (𝑅 “ (𝑅𝐴)))
1918ssriv 3174 1 (dom 𝑅𝐴) ⊆ (𝑅 “ (𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  wa 104  wex 1503  wcel 2160  cin 3143  wss 3144   class class class wbr 4018  ccnv 4643  dom cdm 4644  cima 4647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator