Theorem funssres 5145

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		ssel 3268	. . . . . 6 ⊢ (G ⊆ F → (⟨x, y⟩ ∈ G → ⟨x, y⟩ ∈ F))
2	1	adantl 452	. . . . 5 ⊢ ((Fun F ∧ G ⊆ F) → (⟨x, y⟩ ∈ G → ⟨x, y⟩ ∈ F))
3		opeldm 4911	. . . . . 6 ⊢ (⟨x, y⟩ ∈ G → x ∈ dom G)
4	3	a1i 10	. . . . 5 ⊢ ((Fun F ∧ G ⊆ F) → (⟨x, y⟩ ∈ G → x ∈ dom G))
5	2, 4	jcad 519	. . . 4 ⊢ ((Fun F ∧ G ⊆ F) → (⟨x, y⟩ ∈ G → (⟨x, y⟩ ∈ F ∧ x ∈ dom G)))
6		funeu2 5133	. . . . . . . . . . 11 ⊢ ((Fun F ∧ ⟨x, y⟩ ∈ F) → ∃!y⟨x, y⟩ ∈ F)
7		eldm2 4900	. . . . . . . . . . . . 13 ⊢ (x ∈ dom G ↔ ∃y⟨x, y⟩ ∈ G)
8	1	ancrd 537	. . . . . . . . . . . . . 14 ⊢ (G ⊆ F → (⟨x, y⟩ ∈ G → (⟨x, y⟩ ∈ F ∧ ⟨x, y⟩ ∈ G)))
9	8	eximdv 1622	. . . . . . . . . . . . 13 ⊢ (G ⊆ F → (∃y⟨x, y⟩ ∈ G → ∃y(⟨x, y⟩ ∈ F ∧ ⟨x, y⟩ ∈ G)))
10	7, 9	syl5bi 208	. . . . . . . . . . . 12 ⊢ (G ⊆ F → (x ∈ dom G → ∃y(⟨x, y⟩ ∈ F ∧ ⟨x, y⟩ ∈ G)))
11	10	imp 418	. . . . . . . . . . 11 ⊢ ((G ⊆ F ∧ x ∈ dom G) → ∃y(⟨x, y⟩ ∈ F ∧ ⟨x, y⟩ ∈ G))
12		eupick 2267	. . . . . . . . . . 11 ⊢ ((∃!y⟨x, y⟩ ∈ F ∧ ∃y(⟨x, y⟩ ∈ F ∧ ⟨x, y⟩ ∈ G)) → (⟨x, y⟩ ∈ F → ⟨x, y⟩ ∈ G))
13	6, 11, 12	syl2an 463	. . . . . . . . . 10 ⊢ (((Fun F ∧ ⟨x, y⟩ ∈ F) ∧ (G ⊆ F ∧ x ∈ dom G)) → (⟨x, y⟩ ∈ F → ⟨x, y⟩ ∈ G))
14	13	exp43 595	. . . . . . . . 9 ⊢ (Fun F → (⟨x, y⟩ ∈ F → (G ⊆ F → (x ∈ dom G → (⟨x, y⟩ ∈ F → ⟨x, y⟩ ∈ G)))))
15	14	com23 72	. . . . . . . 8 ⊢ (Fun F → (G ⊆ F → (⟨x, y⟩ ∈ F → (x ∈ dom G → (⟨x, y⟩ ∈ F → ⟨x, y⟩ ∈ G)))))
16	15	imp 418	. . . . . . 7 ⊢ ((Fun F ∧ G ⊆ F) → (⟨x, y⟩ ∈ F → (x ∈ dom G → (⟨x, y⟩ ∈ F → ⟨x, y⟩ ∈ G))))
17	16	com34 77	. . . . . 6 ⊢ ((Fun F ∧ G ⊆ F) → (⟨x, y⟩ ∈ F → (⟨x, y⟩ ∈ F → (x ∈ dom G → ⟨x, y⟩ ∈ G))))
18	17	pm2.43d 44	. . . . 5 ⊢ ((Fun F ∧ G ⊆ F) → (⟨x, y⟩ ∈ F → (x ∈ dom G → ⟨x, y⟩ ∈ G)))
19	18	imp3a 420	. . . 4 ⊢ ((Fun F ∧ G ⊆ F) → ((⟨x, y⟩ ∈ F ∧ x ∈ dom G) → ⟨x, y⟩ ∈ G))
20	5, 19	impbid 183	. . 3 ⊢ ((Fun F ∧ G ⊆ F) → (⟨x, y⟩ ∈ G ↔ (⟨x, y⟩ ∈ F ∧ x ∈ dom G)))
21		opelres 4951	. . 3 ⊢ (⟨x, y⟩ ∈ (F ↾ dom G) ↔ (⟨x, y⟩ ∈ F ∧ x ∈ dom G))
22	20, 21	syl6rbbr 255	. 2 ⊢ ((Fun F ∧ G ⊆ F) → (⟨x, y⟩ ∈ (F ↾ dom G) ↔ ⟨x, y⟩ ∈ G))
23	22	eqrelrdv 4853	1 ⊢ ((Fun F ∧ G ⊆ F) → (F ↾ dom G) = G)

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204   ⊆ wss 3258  ⟨cop 4562  dom cdm 4773   ↾ cres 4775  Fun wfun 4776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790

This theorem is referenced by:  fun2ssres  5146  funcnvres  5166  funssfv  5344  oprssov  5604