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Theorem funssres 4942
 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 2939 . . . . . . 7 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
2 vex 2560 . . . . . . . . 9 𝑥 ∈ V
3 vex 2560 . . . . . . . . 9 𝑦 ∈ V
42, 3opeldm 4538 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐺𝑥 ∈ dom 𝐺)
54a1i 9 . . . . . . 7 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺𝑥 ∈ dom 𝐺))
61, 5jcad 291 . . . . . 6 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺)))
76adantl 262 . . . . 5 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺)))
8 funeu2 4927 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ∃!𝑦𝑥, 𝑦⟩ ∈ 𝐹)
92eldm2 4533 . . . . . . . . . . . . . 14 (𝑥 ∈ dom 𝐺 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐺)
101ancrd 309 . . . . . . . . . . . . . . 15 (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
1110eximdv 1760 . . . . . . . . . . . . . 14 (𝐺𝐹 → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
129, 11syl5bi 141 . . . . . . . . . . . . 13 (𝐺𝐹 → (𝑥 ∈ dom 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
1312imp 115 . . . . . . . . . . . 12 ((𝐺𝐹𝑥 ∈ dom 𝐺) → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
14 eupick 1979 . . . . . . . . . . . 12 ((∃!𝑦𝑥, 𝑦⟩ ∈ 𝐹 ∧ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
158, 13, 14syl2an 273 . . . . . . . . . . 11 (((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺𝐹𝑥 ∈ dom 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
1615exp43 354 . . . . . . . . . 10 (Fun 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝐺𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
1716com23 72 . . . . . . . . 9 (Fun 𝐹 → (𝐺𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
1817imp 115 . . . . . . . 8 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
1918com34 77 . . . . . . 7 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
2019pm2.43d 44 . . . . . 6 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
2120impd 242 . . . . 5 ((Fun 𝐹𝐺𝐹) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺) → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
227, 21impbid 120 . . . 4 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺)))
233opelres 4617 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐺))
2422, 23syl6rbbr 188 . . 3 ((Fun 𝐹𝐺𝐹) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
2524alrimivv 1755 . 2 ((Fun 𝐹𝐺𝐹) → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
26 relres 4639 . . 3 Rel (𝐹 ↾ dom 𝐺)
27 funrel 4919 . . . 4 (Fun 𝐹 → Rel 𝐹)
28 relss 4427 . . . 4 (𝐺𝐹 → (Rel 𝐹 → Rel 𝐺))
2927, 28mpan9 265 . . 3 ((Fun 𝐹𝐺𝐹) → Rel 𝐺)
30 eqrel 4429 . . 3 ((Rel (𝐹 ↾ dom 𝐺) ∧ Rel 𝐺) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
3126, 29, 30sylancr 393 . 2 ((Fun 𝐹𝐺𝐹) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
3225, 31mpbird 156 1 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃!weu 1900   ⊆ wss 2917  ⟨cop 3378  dom cdm 4345   ↾ cres 4347  Rel wrel 4350  Fun wfun 4896 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-fun 4904 This theorem is referenced by:  fun2ssres  4943  funcnvres  4972  funssfv  5199  oprssov  5642
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