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 3463	. . . . 5 ⊢ 𝑦 ∈ V
2	1	opelresi 5974	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
3		vex 3463	. . . . . . . . 9 ⊢ 𝑥 ∈ V
4	3, 1	opeldm 5887	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺)
5	4	a1i 11	. . . . . . 7 ⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺))
6		ssel 3952	. . . . . . 7 ⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
7	5, 6	jcad 512	. . . . . 6 ⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
8	7	adantl 481	. . . . 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 5881	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ dom 𝐺 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐺)
11	6	ancrd 551	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
12	11	eximdv 1917	. . . . . . . . . . . . . 14 ⊢ (𝐺 ⊆ 𝐹 → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
13	10, 12	biimtrid 242	. . . . . . . . . . . . 13 ⊢ (𝐺 ⊆ 𝐹 → (𝑥 ∈ dom 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
14	13	imp 406	. . . . . . . . . . . 12 ⊢ ((𝐺 ⊆ 𝐹 ∧ 𝑥 ∈ dom 𝐺) → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
15		eupick 2632	. . . . . . . . . . . 12 ⊢ ((∃!𝑦⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
16	9, 14, 15	syl2an 596	. . . . . . . . . . 11 ⊢ (((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺 ⊆ 𝐹 ∧ 𝑥 ∈ dom 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
17	16	exp43 436	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝐺 ⊆ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
18	17	com23 86	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
19	18	imp 406	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
20	19	com34 91	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
21	20	pm2.43d 53	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
22	21	impcomd 411	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
23	8, 22	impbid 212	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ↔ (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
24	2, 23	bitr4id 290	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
25	24	alrimivv 1928	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
26		relres 5992	. . 3 ⊢ Rel (𝐹 ↾ dom 𝐺)
27		funrel 6553	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
28		relss 5760	. . . 4 ⊢ (𝐺 ⊆ 𝐹 → (Rel 𝐹 → Rel 𝐺))
29	27, 28	mpan9 506	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → Rel 𝐺)
30		eqrel 5763	. . 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 587	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
32	25, 31	mpbird 257	1 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃!weu 2567   ⊆ wss 3926  ⟨cop 4607  dom cdm 5654   ↾ cres 5656  Rel wrel 5659  Fun wfun 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-res 5666  df-fun 6533

This theorem is referenced by:  fun2ssres  6581  funcnvres  6614  f1ssf1  6850  funssfv  6897  oprssov  7576  isngp2  24536  dvres3  25866  dvres3a  25867  dchrelbas2  27200  issubgr2  29251  uhgrissubgr  29254  funpsstri  35783  funsseq  35785  eqresfnbd  42283