Theorem funssres 6580

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 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)

Proof of Theorem funssres

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3459	. . . . 5 ⊢ 𝑦 ∈ V
2	1	opelresi 5986	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
3		vex 3459	. . . . . . . . 9 ⊢ 𝑥 ∈ V
4	3, 1	opeldm 5897	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺)
5	4	a1i 11	. . . . . . 7 ⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺))
6		ssel 3931	. . . . . . 7 ⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
7	5, 6	jcad 521	. . . . . 6 ⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
8	7	adantl 486	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
9		funeu2 6562	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ∃!𝑦⟨𝑥, 𝑦⟩ ∈ 𝐹)
10	3	eldm2 5891	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ dom 𝐺 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐺)
11	6	ancrd 560	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
12	11	eximdv 1947	. . . . . . . . . . . . . 14 ⊢ (𝐺 ⊆ 𝐹 → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
13	10, 12	biimtrid 245	. . . . . . . . . . . . 13 ⊢ (𝐺 ⊆ 𝐹 → (𝑥 ∈ dom 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
14	13	imp 411	. . . . . . . . . . . 12 ⊢ ((𝐺 ⊆ 𝐹 ∧ 𝑥 ∈ dom 𝐺) → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
15		eupick 2661	. . . . . . . . . . . 12 ⊢ ((∃!𝑦⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
16	9, 14, 15	syl2an 607	. . . . . . . . . . 11 ⊢ (((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺 ⊆ 𝐹 ∧ 𝑥 ∈ dom 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
17	16	exp43 441	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝐺 ⊆ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
18	17	com23 87	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
19	18	imp 411	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
20	19	com34 92	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
21	20	pm2.43d 54	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
22	21	impcomd 416	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
23	8, 22	impbid 215	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ↔ (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
24	2, 23	bitr4id 293	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
25	24	alrimivv 1958	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
26		relres 6004	. . 3 ⊢ Rel (𝐹 ↾ dom 𝐺)
27		funrel 6553	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
28		relss 5768	. . . 4 ⊢ (𝐺 ⊆ 𝐹 → (Rel 𝐹 → Rel 𝐺))
29	27, 28	mpan9 515	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → Rel 𝐺)
30		eqrel 5770	. . 3 ⊢ ((Rel (𝐹 ↾ dom 𝐺) ∧ Rel 𝐺) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
31	26, 29, 30	sylancr 598	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
32	25, 31	mpbird 260	1 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃!weu 2596   ⊆ wss 3905  ⟨cop 4595  dom cdm 5661   ↾ cres 5663  Rel wrel 5666  Fun wfun 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-fun 6538

This theorem is referenced by:  fun2ssres  6581  funcnvres  6614  f1ssf1  6853  funssfv  6902  oprssov  7579  isngp2  24754  dvres3  26072  dvres3a  26073  dchrelbas2  27401  issubgr2  29622  uhgrissubgr  29625  funpsstri  36258  funsseq  36260  eqresfnbd  43003