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Theorem dminss 5997
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 2182 . . . . . . 7 ((𝑥𝐴𝑥𝑅𝑦) → ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
21ancoms 462 . . . . . 6 ((𝑥𝑅𝑦𝑥𝐴) → ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
3 vex 3483 . . . . . . 7 𝑦 ∈ V
43elima2 5922 . . . . . 6 (𝑦 ∈ (𝑅𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
52, 4sylibr 237 . . . . 5 ((𝑥𝑅𝑦𝑥𝐴) → 𝑦 ∈ (𝑅𝐴))
6 simpl 486 . . . . . 6 ((𝑥𝑅𝑦𝑥𝐴) → 𝑥𝑅𝑦)
7 vex 3483 . . . . . . 7 𝑥 ∈ V
83, 7brcnv 5740 . . . . . 6 (𝑦𝑅𝑥𝑥𝑅𝑦)
96, 8sylibr 237 . . . . 5 ((𝑥𝑅𝑦𝑥𝐴) → 𝑦𝑅𝑥)
105, 9jca 515 . . . 4 ((𝑥𝑅𝑦𝑥𝐴) → (𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
1110eximi 1836 . . 3 (∃𝑦(𝑥𝑅𝑦𝑥𝐴) → ∃𝑦(𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
127eldm 5756 . . . . 5 (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
1312anbi1i 626 . . . 4 ((𝑥 ∈ dom 𝑅𝑥𝐴) ↔ (∃𝑦 𝑥𝑅𝑦𝑥𝐴))
14 elin 3935 . . . 4 (𝑥 ∈ (dom 𝑅𝐴) ↔ (𝑥 ∈ dom 𝑅𝑥𝐴))
15 19.41v 1951 . . . 4 (∃𝑦(𝑥𝑅𝑦𝑥𝐴) ↔ (∃𝑦 𝑥𝑅𝑦𝑥𝐴))
1613, 14, 153bitr4i 306 . . 3 (𝑥 ∈ (dom 𝑅𝐴) ↔ ∃𝑦(𝑥𝑅𝑦𝑥𝐴))
177elima2 5922 . . 3 (𝑥 ∈ (𝑅 “ (𝑅𝐴)) ↔ ∃𝑦(𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝑅𝑥))
1811, 16, 173imtr4i 295 . 2 (𝑥 ∈ (dom 𝑅𝐴) → 𝑥 ∈ (𝑅 “ (𝑅𝐴)))
1918ssriv 3957 1 (dom 𝑅𝐴) ⊆ (𝑅 “ (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wa 399  wex 1781  wcel 2115  cin 3918  wss 3919   class class class wbr 5052  ccnv 5541  dom cdm 5542  cima 5545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555
This theorem is referenced by:  lmhmlsp  19821  cnclsi  21880  kgencn3  22166  kqsat  22339  kqcldsat  22341  cfilucfil  23169
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