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

Theorem funssres 6593
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 3479 . . . . 5 𝑦 ∈ V
21opelresi 5990 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
3 vex 3479 . . . . . . . . 9 𝑥 ∈ V
43, 1opeldm 5908 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐺𝑥 ∈ dom 𝐺)
54a1i 11 . . . . . . 7 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺𝑥 ∈ dom 𝐺))
6 ssel 3976 . . . . . . 7 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
75, 6jcad 514 . . . . . 6 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
87adantl 483 . . . . 5 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
9 funeu2 6575 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ∃!𝑦𝑥, 𝑦⟩ ∈ 𝐹)
103eldm2 5902 . . . . . . . . . . . . . 14 (𝑥 ∈ dom 𝐺 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐺)
116ancrd 553 . . . . . . . . . . . . . . 15 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
1211eximdv 1921 . . . . . . . . . . . . . 14 (𝐺𝐹 → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
1310, 12biimtrid 241 . . . . . . . . . . . . 13 (𝐺𝐹 → (𝑥 ∈ dom 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
1413imp 408 . . . . . . . . . . . 12 ((𝐺𝐹𝑥 ∈ dom 𝐺) → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
15 eupick 2630 . . . . . . . . . . . 12 ((∃!𝑦𝑥, 𝑦⟩ ∈ 𝐹 ∧ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
169, 14, 15syl2an 597 . . . . . . . . . . 11 (((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺𝐹𝑥 ∈ dom 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
1716exp43 438 . . . . . . . . . 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 408 . . . . . . . 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 413 . . . . 5 ((Fun 𝐹𝐺𝐹) → ((𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
238, 22impbid 211 . . . 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 1932 . 2 ((Fun 𝐹𝐺𝐹) → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
26 relres 6011 . . 3 Rel (𝐹 ↾ dom 𝐺)
27 funrel 6566 . . . 4 (Fun 𝐹 → Rel 𝐹)
28 relss 5782 . . . 4 (𝐺𝐹 → (Rel 𝐹 → Rel 𝐺))
2927, 28mpan9 508 . . 3 ((Fun 𝐹𝐺𝐹) → Rel 𝐺)
30 eqrel 5785 . . 3 ((Rel (𝐹 ↾ dom 𝐺) ∧ Rel 𝐺) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
3126, 29, 30sylancr 588 . 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 205  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  ∃!weu 2563  wss 3949  cop 4635  dom cdm 5677  cres 5679  Rel wrel 5682  Fun wfun 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-fun 6546
This theorem is referenced by:  fun2ssres  6594  funcnvres  6627  f1ssf1  6866  funssfv  6913  oprssov  7576  isngp2  24106  dvres3  25430  dvres3a  25431  dchrelbas2  26740  issubgr2  28529  uhgrissubgr  28532  funpsstri  34737  funsseq  34739  eqresfnbd  41054
  Copyright terms: Public domain W3C validator