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

Theorem funssres 6612
Description: The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.)
Assertion
Ref Expression
funssres ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)

Proof of Theorem funssres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3482 . . . . 5 𝑦 ∈ V
21opelresi 6008 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
3 vex 3482 . . . . . . . . 9 𝑥 ∈ V
43, 1opeldm 5921 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐺𝑥 ∈ dom 𝐺)
54a1i 11 . . . . . . 7 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺𝑥 ∈ dom 𝐺))
6 ssel 3989 . . . . . . 7 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
75, 6jcad 512 . . . . . 6 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
87adantl 481 . . . . 5 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
9 funeu2 6594 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ∃!𝑦𝑥, 𝑦⟩ ∈ 𝐹)
103eldm2 5915 . . . . . . . . . . . . . 14 (𝑥 ∈ dom 𝐺 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐺)
116ancrd 551 . . . . . . . . . . . . . . 15 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
1211eximdv 1915 . . . . . . . . . . . . . 14 (𝐺𝐹 → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
1310, 12biimtrid 242 . . . . . . . . . . . . 13 (𝐺𝐹 → (𝑥 ∈ dom 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
1413imp 406 . . . . . . . . . . . 12 ((𝐺𝐹𝑥 ∈ dom 𝐺) → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
15 eupick 2631 . . . . . . . . . . . 12 ((∃!𝑦𝑥, 𝑦⟩ ∈ 𝐹 ∧ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
169, 14, 15syl2an 596 . . . . . . . . . . 11 (((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺𝐹𝑥 ∈ dom 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
1716exp43 436 . . . . . . . . . 10 (Fun 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝐺𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
1817com23 86 . . . . . . . . 9 (Fun 𝐹 → (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
1918imp 406 . . . . . . . 8 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
2019com34 91 . . . . . . 7 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
2120pm2.43d 53 . . . . . 6 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
2221impcomd 411 . . . . 5 ((Fun 𝐹𝐺𝐹) → ((𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
238, 22impbid 212 . . . 4 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ↔ (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
242, 23bitr4id 290 . . 3 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
2524alrimivv 1926 . 2 ((Fun 𝐹𝐺𝐹) → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
26 relres 6026 . . 3 Rel (𝐹 ↾ dom 𝐺)
27 funrel 6585 . . . 4 (Fun 𝐹 → Rel 𝐹)
28 relss 5794 . . . 4 (𝐺𝐹 → (Rel 𝐹 → Rel 𝐺))
2927, 28mpan9 506 . . 3 ((Fun 𝐹𝐺𝐹) → Rel 𝐺)
30 eqrel 5797 . . 3 ((Rel (𝐹 ↾ dom 𝐺) ∧ Rel 𝐺) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
3126, 29, 30sylancr 587 . 2 ((Fun 𝐹𝐺𝐹) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
3225, 31mpbird 257 1 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  ∃!weu 2566  wss 3963  cop 4637  dom cdm 5689  cres 5691  Rel wrel 5694  Fun wfun 6557
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-mo 2538  df-eu 2567  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-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-fun 6565
This theorem is referenced by:  fun2ssres  6613  funcnvres  6646  f1ssf1  6881  funssfv  6928  oprssov  7602  isngp2  24626  dvres3  25963  dvres3a  25964  dchrelbas2  27296  issubgr2  29304  uhgrissubgr  29307  funpsstri  35747  funsseq  35749  eqresfnbd  42252
  Copyright terms: Public domain W3C validator