MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dminss Structured version   Visualization version   GIF version

Theorem dminss 6145
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 2166 . . . . . . 7 ((𝑥𝐴𝑥𝑅𝑦) → ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
21ancoms 458 . . . . . 6 ((𝑥𝑅𝑦𝑥𝐴) → ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
3 vex 3472 . . . . . . 7 𝑦 ∈ V
43elima2 6058 . . . . . 6 (𝑦 ∈ (𝑅𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
52, 4sylibr 233 . . . . 5 ((𝑥𝑅𝑦𝑥𝐴) → 𝑦 ∈ (𝑅𝐴))
6 simpl 482 . . . . . 6 ((𝑥𝑅𝑦𝑥𝐴) → 𝑥𝑅𝑦)
7 vex 3472 . . . . . . 7 𝑥 ∈ V
83, 7brcnv 5875 . . . . . 6 (𝑦𝑅𝑥𝑥𝑅𝑦)
96, 8sylibr 233 . . . . 5 ((𝑥𝑅𝑦𝑥𝐴) → 𝑦𝑅𝑥)
105, 9jca 511 . . . 4 ((𝑥𝑅𝑦𝑥𝐴) → (𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
1110eximi 1829 . . 3 (∃𝑦(𝑥𝑅𝑦𝑥𝐴) → ∃𝑦(𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
127eldm 5893 . . . . 5 (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
1312anbi1i 623 . . . 4 ((𝑥 ∈ dom 𝑅𝑥𝐴) ↔ (∃𝑦 𝑥𝑅𝑦𝑥𝐴))
14 elin 3959 . . . 4 (𝑥 ∈ (dom 𝑅𝐴) ↔ (𝑥 ∈ dom 𝑅𝑥𝐴))
15 19.41v 1945 . . . 4 (∃𝑦(𝑥𝑅𝑦𝑥𝐴) ↔ (∃𝑦 𝑥𝑅𝑦𝑥𝐴))
1613, 14, 153bitr4i 303 . . 3 (𝑥 ∈ (dom 𝑅𝐴) ↔ ∃𝑦(𝑥𝑅𝑦𝑥𝐴))
177elima2 6058 . . 3 (𝑥 ∈ (𝑅 “ (𝑅𝐴)) ↔ ∃𝑦(𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
1811, 16, 173imtr4i 292 . 2 (𝑥 ∈ (dom 𝑅𝐴) → 𝑥 ∈ (𝑅 “ (𝑅𝐴)))
1918ssriv 3981 1 (dom 𝑅𝐴) ⊆ (𝑅 “ (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wa 395  wex 1773  wcel 2098  cin 3942  wss 3943   class class class wbr 5141  ccnv 5668  dom cdm 5669  cima 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682
This theorem is referenced by:  lmhmlsp  20895  cnclsi  23127  kgencn3  23413  kqsat  23586  kqcldsat  23588  cfilucfil  24419  elrspunidl  33052
  Copyright terms: Public domain W3C validator