Theorem funssres 6393

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 3498	. . . . . . . . 9 ⊢ 𝑥 ∈ V
2		vex 3498	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3	1, 2	opeldm 5771	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺)
4	3	a1i 11	. . . . . . 7 ⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺))
5		ssel 3961	. . . . . . 7 ⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
6	4, 5	jcad 515	. . . . . 6 ⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
7	6	adantl 484	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
8		funeu2 6376	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ∃!𝑦⟨𝑥, 𝑦⟩ ∈ 𝐹)
9	1	eldm2 5765	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ dom 𝐺 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐺)
10	5	ancrd 554	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
11	10	eximdv 1914	. . . . . . . . . . . . . 14 ⊢ (𝐺 ⊆ 𝐹 → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
12	9, 11	syl5bi 244	. . . . . . . . . . . . 13 ⊢ (𝐺 ⊆ 𝐹 → (𝑥 ∈ dom 𝐺 → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
13	12	imp 409	. . . . . . . . . . . 12 ⊢ ((𝐺 ⊆ 𝐹 ∧ 𝑥 ∈ dom 𝐺) → ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
14		eupick 2714	. . . . . . . . . . . 12 ⊢ ((∃!𝑦⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
15	8, 13, 14	syl2an 597	. . . . . . . . . . 11 ⊢ (((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺 ⊆ 𝐹 ∧ 𝑥 ∈ dom 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
16	15	exp43 439	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝐺 ⊆ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
17	16	com23 86	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))))
18	17	imp 409	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
19	18	com34 91	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺))))
20	19	pm2.43d 53	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ dom 𝐺 → ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
21	20	impcomd 414	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ⟨𝑥, 𝑦⟩ ∈ 𝐺))
22	7, 21	impbid 214	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ↔ (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
23	2	opelresi 5856	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
24	22, 23	syl6rbbr 292	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
25	24	alrimivv 1925	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
26		relres 5877	. . 3 ⊢ Rel (𝐹 ↾ dom 𝐺)
27		funrel 6367	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
28		relss 5651	. . . 4 ⊢ (𝐺 ⊆ 𝐹 → (Rel 𝐹 → Rel 𝐺))
29	27, 28	mpan9 509	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → Rel 𝐺)
30		eqrel 5653	. . 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 589	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝐹 ↾ dom 𝐺) = 𝐺 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)))
32	25, 31	mpbird 259	1 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∃!weu 2649   ⊆ wss 3936  ⟨cop 4567  dom cdm 5550   ↾ cres 5552  Rel wrel 5555  Fun wfun 6344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-res 5562  df-fun 6352

This theorem is referenced by:  fun2ssres  6394  funcnvres  6427  f1ssf1  6641  funssfv  6686  oprssov  7311  isngp2  23200  dvres3  24505  dvres3a  24506  dchrelbas2  25807  issubgr2  27048  uhgrissubgr  27051  funpsstri  33003  funsseq  33006