Theorem fvsnun2 5382

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5381. (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 4626	. . . 4 ⊢ (𝐺 ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴}))
3		resundir 4644	. . . 4 ⊢ (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})))
4		disjdif 3316	. . . . . . 7 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅
5		fvsnun.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
6		fvsnun.2	. . . . . . . . 9 ⊢ 𝐵 ∈ V
7	5, 6	fnsn 4973	. . . . . . . 8 ⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴}
8		fnresdisj 5029	. . . . . . . 8 ⊢ ({⟨𝐴, 𝐵⟩} Fn {𝐴} → (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅))
9	7, 8	ax-mp 7	. . . . . . 7 ⊢ (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅)
10	4, 9	mpbi 143	. . . . . 6 ⊢ ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅
11		residm 4660	. . . . . 6 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
12	10, 11	uneq12i 3124	. . . . 5 ⊢ (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
13		uncom 3116	. . . . 5 ⊢ (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅)
14		un0 3278	. . . . 5 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
15	12, 13, 14	3eqtri 2105	. . . 4 ⊢ (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
16	2, 3, 15	3eqtri 2105	. . 3 ⊢ (𝐺 ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
17	16	fveq1i 5199	. 2 ⊢ ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷)
18		fvres 5219	. 2 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐺‘𝐷))
19		fvres 5219	. 2 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷))
20	17, 18, 19	3eqtr3a 2137	1 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → (𝐺‘𝐷) = (𝐹‘𝐷))

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ∖ cdif 2970   ∪ cun 2971   ∩ cin 2972  ∅c0 3251  {csn 3398  ⟨cop 3401   ↾ cres 4365   Fn wfn 4917  ‘cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-res 4375  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by:  facnn  9654