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Theorem funssres 6089
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 ssel 3736 . . . . . . 7 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
2 vex 3341 . . . . . . . . 9 𝑥 ∈ V
3 vex 3341 . . . . . . . . 9 𝑦 ∈ V
42, 3opeldm 5481 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐺𝑥 ∈ dom 𝐺)
54a1i 11 . . . . . . 7 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺𝑥 ∈ dom 𝐺))
61, 5jcad 556 . . . . . 6 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺)))
76adantl 473 . . . . 5 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺)))
8 funeu2 6073 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ∃!𝑦𝑥, 𝑦⟩ ∈ 𝐹)
92eldm2 5475 . . . . . . . . . . . . . 14 (𝑥 ∈ dom 𝐺 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐺)
101ancrd 578 . . . . . . . . . . . . . . 15 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
1110eximdv 1993 . . . . . . . . . . . . . 14 (𝐺𝐹 → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
129, 11syl5bi 232 . . . . . . . . . . . . 13 (𝐺𝐹 → (𝑥 ∈ dom 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
1312imp 444 . . . . . . . . . . . 12 ((𝐺𝐹𝑥 ∈ dom 𝐺) → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
14 eupick 2672 . . . . . . . . . . . 12 ((∃!𝑦𝑥, 𝑦⟩ ∈ 𝐹 ∧ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
158, 13, 14syl2an 495 . . . . . . . . . . 11 (((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺𝐹𝑥 ∈ dom 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
1615exp43 641 . . . . . . . . . 10 (Fun 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝐺𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
1716com23 86 . . . . . . . . 9 (Fun 𝐹 → (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
1817imp 444 . . . . . . . 8 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
1918com34 91 . . . . . . 7 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
2019pm2.43d 53 . . . . . 6 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
2120impd 446 . . . . 5 ((Fun 𝐹𝐺𝐹) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺) → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
227, 21impbid 202 . . . 4 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺)))
233opelres 5557 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺))
2422, 23syl6rbbr 279 . . 3 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
2524alrimivv 2003 . 2 ((Fun 𝐹𝐺𝐹) → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
26 relres 5582 . . 3 Rel (𝐹 ↾ dom 𝐺)
27 funrel 6064 . . . 4 (Fun 𝐹 → Rel 𝐹)
28 relss 5361 . . . 4 (𝐺𝐹 → (Rel 𝐹 → Rel 𝐺))
2927, 28mpan9 487 . . 3 ((Fun 𝐹𝐺𝐹) → Rel 𝐺)
30 eqrel 5364 . . 3 ((Rel (𝐹 ↾ dom 𝐺) ∧ Rel 𝐺) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
3126, 29, 30sylancr 698 . 2 ((Fun 𝐹𝐺𝐹) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
3225, 31mpbird 247 1 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1628   = wceq 1630  wex 1851  wcel 2137  ∃!weu 2605  wss 3713  cop 4325  dom cdm 5264  cres 5266  Rel wrel 5269  Fun wfun 6041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-res 5276  df-fun 6049
This theorem is referenced by:  fun2ssres  6090  funcnvres  6126  f1ssf1  6327  funssfv  6368  oprssov  6966  isngp2  22600  dvres3  23874  dvres3a  23875  dchrelbas2  25159  issubgr2  26361  uhgrissubgr  26364  funpsstri  31968  funsseq  31971
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