Theorem fvsnun2 5448

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5447. (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 5342	. 2 ⊢ (D ∈ (C ∖ {A}) → ((G ↾ (C ∖ {A})) ‘D) = (G ‘D))
2		fvsnun.3	. . . . . 6 ⊢ G = ({⟨A, B⟩} ∪ (F ↾ (C ∖ {A})))
3	2	reseq1i 4930	. . . . 5 ⊢ (G ↾ (C ∖ {A})) = (({⟨A, B⟩} ∪ (F ↾ (C ∖ {A}))) ↾ (C ∖ {A}))
4		resundir 4982	. . . . 5 ⊢ (({⟨A, B⟩} ∪ (F ↾ (C ∖ {A}))) ↾ (C ∖ {A})) = (({⟨A, B⟩} ↾ (C ∖ {A})) ∪ ((F ↾ (C ∖ {A})) ↾ (C ∖ {A})))
5		disjdif 3622	. . . . . . . 8 ⊢ ({A} ∩ (C ∖ {A})) = ∅
6		fvsnun.1	. . . . . . . . . 10 ⊢ A ∈ V
7		fvsnun.2	. . . . . . . . . 10 ⊢ B ∈ V
8	6, 7	fnsn 5152	. . . . . . . . 9 ⊢ {⟨A, B⟩} Fn {A}
9		fnresdisj 5193	. . . . . . . . 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 4994	. . . . . . 7 ⊢ ((F ↾ (C ∖ {A})) ↾ (C ∖ {A})) = (F ↾ (C ∖ {A}))
13	11, 12	uneq12i 3416	. . . . . 6 ⊢ (({⟨A, B⟩} ↾ (C ∖ {A})) ∪ ((F ↾ (C ∖ {A})) ↾ (C ∖ {A}))) = (∅ ∪ (F ↾ (C ∖ {A})))
14		uncom 3408	. . . . . 6 ⊢ (∅ ∪ (F ↾ (C ∖ {A}))) = ((F ↾ (C ∖ {A})) ∪ ∅)
15		un0 3575	. . . . . 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 5329	. . 3 ⊢ ((G ↾ (C ∖ {A})) ‘D) = ((F ↾ (C ∖ {A})) ‘D)
19		fvres 5342	. . 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 2859   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  ⟨cop 4561   ↾ cres 4774   Fn wfn 4776   ‘cfv 4781

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-fv 4795

This theorem is referenced by: (None)