Theorem fvsnun1 5850

Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5851. (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
fvsnun1	⊢ (𝐺‘𝐴) = 𝐵

Proof of Theorem fvsnun1

Step	Hyp	Ref	Expression
1		fvsnun.3	. . . . 5 ⊢ 𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
2	1	reseq1i 5009	. . . 4 ⊢ (𝐺 ↾ {𝐴}) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴})
3		resundir 5027	. . . . 5 ⊢ (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}))
4		incom 3399	. . . . . . . . 9 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ({𝐴} ∩ (𝐶 ∖ {𝐴}))
5		disjdif 3567	. . . . . . . . 9 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅
6	4, 5	eqtri 2252	. . . . . . . 8 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅
7		resdisj 5165	. . . . . . . 8 ⊢ (((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅)
8	6, 7	ax-mp 5	. . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅
9	8	uneq2i 3358	. . . . . 6 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ∅)
10		un0 3528	. . . . . 6 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ∅) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
11	9, 10	eqtri 2252	. . . . 5 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
12	3, 11	eqtri 2252	. . . 4 ⊢ (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
13	2, 12	eqtri 2252	. . 3 ⊢ (𝐺 ↾ {𝐴}) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
14	13	fveq1i 5640	. 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴)
15		fvsnun.1	. . . 4 ⊢ 𝐴 ∈ V
16	15	snid 3700	. . 3 ⊢ 𝐴 ∈ {𝐴}
17		fvres 5663	. . 3 ⊢ (𝐴 ∈ {𝐴} → ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴))
18	16, 17	ax-mp 5	. 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴)
19		fvres 5663	. . . 4 ⊢ (𝐴 ∈ {𝐴} → (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = ({⟨𝐴, 𝐵⟩}‘𝐴))
20	16, 19	ax-mp 5	. . 3 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = ({⟨𝐴, 𝐵⟩}‘𝐴)
21		fvsnun.2	. . . 4 ⊢ 𝐵 ∈ V
22	15, 21	fvsn 5848	. . 3 ⊢ ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵
23	20, 22	eqtri 2252	. 2 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = 𝐵
24	14, 18, 23	3eqtr3i 2260	1 ⊢ (𝐺‘𝐴) = 𝐵

Syntax hints:   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ∖ cdif 3197   ∪ cun 3198   ∩ cin 3199  ∅c0 3494  {csn 3669  ⟨cop 3672   ↾ cres 4727  ‘cfv 5326

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-res 4737  df-iota 5286  df-fun 5328  df-fv 5334

This theorem is referenced by:  fac0  10989