Theorem funssres 6533

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 3441	. . . . 5 ⊢ 𝑦 ∈ V
2	1	opelresi 5943	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ (𝑥 ∈ dom 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
3		vex 3441	. . . . . . . . 9 ⊢ 𝑥 ∈ V
4	3, 1	opeldm 5853	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺)
5	4	a1i 11	. . . . . . 7 ⊢ (𝐺 ⊆ 𝐹 → (⟨𝑥, 𝑦⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺))
6		ssel 3924	. . . . . . 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 6515	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ∃!𝑦⟨𝑥, 𝑦⟩ ∈ 𝐹)
10	3	eldm2 5847	. . . . . . . . . . . . . 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 1918	. . . . . . . . . . . . . 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 2630	. . . . . . . . . . . 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 1929	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ dom 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
26		relres 5961	. . 3 ⊢ Rel (𝐹 ↾ dom 𝐺)
27		funrel 6506	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
28		relss 5728	. . . 4 ⊢ (𝐺 ⊆ 𝐹 → (Rel 𝐹 → Rel 𝐺))
29	27, 28	mpan9 506	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → Rel 𝐺)
30		eqrel 5730	. . 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 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃!weu 2565   ⊆ wss 3898  ⟨cop 4583  dom cdm 5621   ↾ cres 5623  Rel wrel 5626  Fun wfun 6483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-res 5633  df-fun 6491

This theorem is referenced by:  fun2ssres  6534  funcnvres  6567  f1ssf1  6803  funssfv  6852  oprssov  7524  isngp2  24532  dvres3  25861  dvres3a  25862  dchrelbas2  27195  issubgr2  29271  uhgrissubgr  29274  funpsstri  35882  funsseq  35884  eqresfnbd  42403