Theorem fvsnun2 5781

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5780. (Contributed by NM, 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		fvsnun.3	. . . . 5 ⊢ 𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
2	1	reseq1i 4954	. . . 4 ⊢ (𝐺 ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴}))
3		resundir 4972	. . . 4 ⊢ (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})))
4		disjdif 3532	. . . . . . 7 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅
5		fvsnun.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
6		fvsnun.2	. . . . . . . . 9 ⊢ 𝐵 ∈ V
7	5, 6	fnsn 5327	. . . . . . . 8 ⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴}
8		fnresdisj 5385	. . . . . . . 8 ⊢ ({⟨𝐴, 𝐵⟩} Fn {𝐴} → (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅))
9	7, 8	ax-mp 5	. . . . . . 7 ⊢ (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅)
10	4, 9	mpbi 145	. . . . . 6 ⊢ ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅
11		residm 4990	. . . . . 6 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
12	10, 11	uneq12i 3324	. . . . 5 ⊢ (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
13		uncom 3316	. . . . 5 ⊢ (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅)
14		un0 3493	. . . . 5 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
15	12, 13, 14	3eqtri 2229	. . . 4 ⊢ (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
16	2, 3, 15	3eqtri 2229	. . 3 ⊢ (𝐺 ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
17	16	fveq1i 5576	. 2 ⊢ ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷)
18		fvres 5599	. 2 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐺‘𝐷))
19		fvres 5599	. 2 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷))
20	17, 18, 19	3eqtr3a 2261	1 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → (𝐺‘𝐷) = (𝐹‘𝐷))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1372   ∈ wcel 2175  Vcvv 2771   ∖ cdif 3162   ∪ cun 3163   ∩ cin 3164  ∅c0 3459  {csn 3632  ⟨cop 3635   ↾ cres 4676   Fn wfn 5265  ‘cfv 5270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-fv 5278

This theorem is referenced by:  facnn  10870