Theorem fvsnun2 5449

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5448. (Contributed by set.mm contributors, 23-Sep-2007.)

Hypotheses

Ref	Expression
fvsnun.1	⊢ A ∈ V
fvsnun.2	⊢ B ∈ V
fvsnun.3	⊢ G = ({⟨A, B⟩} ∪ (F ↾ (C ∖ {A})))

Assertion

Ref	Expression
fvsnun2	⊢ (D ∈ (C ∖ {A}) → (G ‘D) = (F ‘D))

Proof of Theorem fvsnun2

Step	Hyp	Ref	Expression
1		fvres 5343	. 2 ⊢ (D ∈ (C ∖ {A}) → ((G ↾ (C ∖ {A})) ‘D) = (G ‘D))
2		fvsnun.3	. . . . . 6 ⊢ G = ({⟨A, B⟩} ∪ (F ↾ (C ∖ {A})))
3	2	reseq1i 4931	. . . . 5 ⊢ (G ↾ (C ∖ {A})) = (({⟨A, B⟩} ∪ (F ↾ (C ∖ {A}))) ↾ (C ∖ {A}))
4		resundir 4983	. . . . 5 ⊢ (({⟨A, B⟩} ∪ (F ↾ (C ∖ {A}))) ↾ (C ∖ {A})) = (({⟨A, B⟩} ↾ (C ∖ {A})) ∪ ((F ↾ (C ∖ {A})) ↾ (C ∖ {A})))
5		disjdif 3623	. . . . . . . 8 ⊢ ({A} ∩ (C ∖ {A})) = ∅
6		fvsnun.1	. . . . . . . . . 10 ⊢ A ∈ V
7		fvsnun.2	. . . . . . . . . 10 ⊢ B ∈ V
8	6, 7	fnsn 5153	. . . . . . . . 9 ⊢ {⟨A, B⟩} Fn {A}
9		fnresdisj 5194	. . . . . . . . 9 ⊢ ({⟨A, B⟩} Fn {A} → (({A} ∩ (C ∖ {A})) = ∅ ↔ ({⟨A, B⟩} ↾ (C ∖ {A})) = ∅))
10	8, 9	ax-mp 5	. . . . . . . 8 ⊢ (({A} ∩ (C ∖ {A})) = ∅ ↔ ({⟨A, B⟩} ↾ (C ∖ {A})) = ∅)
11	5, 10	mpbi 199	. . . . . . 7 ⊢ ({⟨A, B⟩} ↾ (C ∖ {A})) = ∅
12		residm 4995	. . . . . . 7 ⊢ ((F ↾ (C ∖ {A})) ↾ (C ∖ {A})) = (F ↾ (C ∖ {A}))
13	11, 12	uneq12i 3417	. . . . . 6 ⊢ (({⟨A, B⟩} ↾ (C ∖ {A})) ∪ ((F ↾ (C ∖ {A})) ↾ (C ∖ {A}))) = (∅ ∪ (F ↾ (C ∖ {A})))
14		uncom 3409	. . . . . 6 ⊢ (∅ ∪ (F ↾ (C ∖ {A}))) = ((F ↾ (C ∖ {A})) ∪ ∅)
15		un0 3576	. . . . . 6 ⊢ ((F ↾ (C ∖ {A})) ∪ ∅) = (F ↾ (C ∖ {A}))
16	13, 14, 15	3eqtri 2377	. . . . 5 ⊢ (({⟨A, B⟩} ↾ (C ∖ {A})) ∪ ((F ↾ (C ∖ {A})) ↾ (C ∖ {A}))) = (F ↾ (C ∖ {A}))
17	3, 4, 16	3eqtri 2377	. . . 4 ⊢ (G ↾ (C ∖ {A})) = (F ↾ (C ∖ {A}))
18	17	fveq1i 5330	. . 3 ⊢ ((G ↾ (C ∖ {A})) ‘D) = ((F ↾ (C ∖ {A})) ‘D)
19		fvres 5343	. . 3 ⊢ (D ∈ (C ∖ {A}) → ((F ↾ (C ∖ {A})) ‘D) = (F ‘D))
20	18, 19	syl5eq 2397	. 2 ⊢ (D ∈ (C ∖ {A}) → ((G ↾ (C ∖ {A})) ‘D) = (F ‘D))
21	1, 20	eqtr3d 2387	1 ⊢ (D ∈ (C ∖ {A}) → (G ‘D) = (F ‘D))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738  ⟨cop 4562   ↾ cres 4775   Fn wfn 4777   ‘cfv 4782

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  df-fn 4791  df-fv 4796

This theorem is referenced by: (None)