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Theorem funssres 5211
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 2715 . . . . 5 𝑦 ∈ V
21opelres 4870 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺))
3 ssel 3122 . . . . . . 7 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
4 vex 2715 . . . . . . . . 9 𝑥 ∈ V
54, 1opeldm 4788 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐺𝑥 ∈ dom 𝐺)
65a1i 9 . . . . . . 7 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺𝑥 ∈ dom 𝐺))
73, 6jcad 305 . . . . . 6 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺)))
87adantl 275 . . . . 5 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺)))
9 funeu2 5195 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ∃!𝑦𝑥, 𝑦⟩ ∈ 𝐹)
104eldm2 4783 . . . . . . . . . . . . . 14 (𝑥 ∈ dom 𝐺 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐺)
113ancrd 324 . . . . . . . . . . . . . . 15 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
1211eximdv 1860 . . . . . . . . . . . . . 14 (𝐺𝐹 → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
1310, 12syl5bi 151 . . . . . . . . . . . . 13 (𝐺𝐹 → (𝑥 ∈ dom 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
1413imp 123 . . . . . . . . . . . 12 ((𝐺𝐹𝑥 ∈ dom 𝐺) → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
15 eupick 2085 . . . . . . . . . . . 12 ((∃!𝑦𝑥, 𝑦⟩ ∈ 𝐹 ∧ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
169, 14, 15syl2an 287 . . . . . . . . . . 11 (((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺𝐹𝑥 ∈ dom 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
1716exp43 370 . . . . . . . . . 10 (Fun 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝐺𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
1817com23 78 . . . . . . . . 9 (Fun 𝐹 → (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
1918imp 123 . . . . . . . 8 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
2019com34 83 . . . . . . 7 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
2120pm2.43d 50 . . . . . 6 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
2221impd 252 . . . . 5 ((Fun 𝐹𝐺𝐹) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺) → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
238, 22impbid 128 . . . 4 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺)))
242, 23bitr4id 198 . . 3 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
2524alrimivv 1855 . 2 ((Fun 𝐹𝐺𝐹) → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
26 relres 4893 . . 3 Rel (𝐹 ↾ dom 𝐺)
27 funrel 5186 . . . 4 (Fun 𝐹 → Rel 𝐹)
28 relss 4672 . . . 4 (𝐺𝐹 → (Rel 𝐹 → Rel 𝐺))
2927, 28mpan9 279 . . 3 ((Fun 𝐹𝐺𝐹) → Rel 𝐺)
30 eqrel 4674 . . 3 ((Rel (𝐹 ↾ dom 𝐺) ∧ Rel 𝐺) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
3126, 29, 30sylancr 411 . 2 ((Fun 𝐹𝐺𝐹) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
3225, 31mpbird 166 1 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1333   = wceq 1335  wex 1472  ∃!weu 2006  wcel 2128  wss 3102  cop 3563  dom cdm 4585  cres 4587  Rel wrel 4590  Fun wfun 5163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-res 4597  df-fun 5171
This theorem is referenced by:  fun2ssres  5212  funcnvres  5242  funssfv  5493  oprssov  5959
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