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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 8	. . . . . . . 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))

Colors of variables: wff setvar class

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-1 5 ax-2 6 ax-3 7 ax-mp 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)

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