Theorem fvsnun2 5836

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5835. (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 5000	. . . 4 ⊢ (𝐺 ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴}))
3		resundir 5018	. . . 4 ⊢ (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})))
4		disjdif 3564	. . . . . . 7 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅
5		fvsnun.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
6		fvsnun.2	. . . . . . . . 9 ⊢ 𝐵 ∈ V
7	5, 6	fnsn 5374	. . . . . . . 8 ⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴}
8		fnresdisj 5432	. . . . . . . 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 5036	. . . . . 6 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
12	10, 11	uneq12i 3356	. . . . 5 ⊢ (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
13		uncom 3348	. . . . 5 ⊢ (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅)
14		un0 3525	. . . . 5 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
15	12, 13, 14	3eqtri 2254	. . . 4 ⊢ (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
16	2, 3, 15	3eqtri 2254	. . 3 ⊢ (𝐺 ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
17	16	fveq1i 5627	. 2 ⊢ ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷)
18		fvres 5650	. 2 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐺‘𝐷))
19		fvres 5650	. 2 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷))
20	17, 18, 19	3eqtr3a 2286	1 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → (𝐺‘𝐷) = (𝐹‘𝐷))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ∖ cdif 3194   ∪ cun 3195   ∩ cin 3196  ∅c0 3491  {csn 3666  ⟨cop 3669   ↾ cres 4720   Fn wfn 5312  ‘cfv 5317

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325

This theorem is referenced by:  facnn  10944