Theorem dffv2 6806

Description: Alternate definition of function value df-fv 6388 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.)

Assertion

Ref	Expression
dffv2	⊢ (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))

Proof of Theorem dffv2

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snidb 4576	. . . . 5 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
2		fvres 6736	. . . . 5 ⊢ (𝐴 ∈ {𝐴} → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴))
3	1, 2	sylbi 220	. . . 4 ⊢ (𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴))
4		fvprc 6709	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = ∅)
5		fvprc 6709	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
6	4, 5	eqtr4d 2780	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴))
7	3, 6	pm2.61i 185	. . 3 ⊢ ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴)
8		funfv 6798	. . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴})‘𝐴) = ∪ ((𝐹 ↾ {𝐴}) “ {𝐴}))
9		resima 5885	. . . . . . 7 ⊢ ((𝐹 ↾ {𝐴}) “ {𝐴}) = (𝐹 “ {𝐴})
10		dif0 4287	. . . . . . 7 ⊢ ((𝐹 “ {𝐴}) ∖ ∅) = (𝐹 “ {𝐴})
11	9, 10	eqtr4i 2768	. . . . . 6 ⊢ ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ ∅)
12		df-fun 6382	. . . . . . . . . . . . 13 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ⊆ I ))
13	12	simprbi 500	. . . . . . . . . . . 12 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ⊆ I )
14		ssdif0 4278	. . . . . . . . . . . 12 ⊢ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ⊆ I ↔ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∅)
15	13, 14	sylib 221	. . . . . . . . . . 11 ⊢ (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∅)
16	15	unieqd 4833	. . . . . . . . . 10 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∪ ∅)
17		uni0 4849	. . . . . . . . . 10 ⊢ ∪ ∅ = ∅
18	16, 17	eqtrdi 2794	. . . . . . . . 9 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∅)
19	18	unieqd 4833	. . . . . . . 8 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∪ ∅)
20	19, 17	eqtrdi 2794	. . . . . . 7 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∅)
21	20	difeq2d 4037	. . . . . 6 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ((𝐹 “ {𝐴}) ∖ ∅))
22	11, 21	eqtr4id 2797	. . . . 5 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
23	22	unieqd 4833	. . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ ((𝐹 ↾ {𝐴}) “ {𝐴}) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
24	8, 23	eqtrd 2777	. . 3 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴})‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
25	7, 24	eqtr3id 2792	. 2 ⊢ (Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
26		nfunsn 6754	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)
27		relres 5880	. . . . . . . . . . . . . . 15 ⊢ Rel (𝐹 ↾ {𝐴})
28		dffun3 6391	. . . . . . . . . . . . . . 15 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦)))
29	27, 28	mpbiran 709	. . . . . . . . . . . . . 14 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ ∀𝑥∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦))
30		iman 405	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
31	30	albii 1827	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧 ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
32		alnex 1789	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑧 ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
33	31, 32	bitri 278	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
34	33	exbii 1855	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ∃𝑦 ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
35		exnal 1834	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑦 ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
36	34, 35	bitri 278	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ¬ ∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
37	36	albii 1827	. . . . . . . . . . . . . 14 ⊢ (∀𝑥∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ∀𝑥 ¬ ∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
38		alnex 1789	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ¬ ∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∃𝑥∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
39	29, 37, 38	3bitrri 301	. . . . . . . . . . . . 13 ⊢ (¬ ∃𝑥∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ Fun (𝐹 ↾ {𝐴}))
40	39	con1bii 360	. . . . . . . . . . . 12 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) ↔ ∃𝑥∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
41		sp 2180	. . . . . . . . . . . . 13 ⊢ (∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
42	41	eximi 1842	. . . . . . . . . . . 12 ⊢ (∃𝑥∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑥∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
43	40, 42	sylbi 220	. . . . . . . . . . 11 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ∃𝑥∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
44		snssi 4721	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → {𝐴} ⊆ dom (𝐹 ↾ {𝐴}))
45		residm 5884	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = (𝐹 ↾ {𝐴})
46	45	dmeqi 5773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = dom (𝐹 ↾ {𝐴})
47		ssdmres 5874	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) ↔ dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = {𝐴})
48	47	biimpi 219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) → dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = {𝐴})
49	46, 48	eqtr3id 2792	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) → dom (𝐹 ↾ {𝐴}) = {𝐴})
50	44, 49	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → dom (𝐹 ↾ {𝐴}) = {𝐴})
51		vex 3412	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
52		vex 3412	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑧 ∈ V
53	51, 52	breldm 5777	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑥 ∈ dom (𝐹 ↾ {𝐴}))
54		eleq2 2826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (dom (𝐹 ↾ {𝐴}) = {𝐴} → (𝑥 ∈ dom (𝐹 ↾ {𝐴}) ↔ 𝑥 ∈ {𝐴}))
55		velsn 4557	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
56	54, 55	bitrdi 290	. . . . . . . . . . . . . . . . . . . 20 ⊢ (dom (𝐹 ↾ {𝐴}) = {𝐴} → (𝑥 ∈ dom (𝐹 ↾ {𝐴}) ↔ 𝑥 = 𝐴))
57	56	biimpa 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom (𝐹 ↾ {𝐴}) = {𝐴} ∧ 𝑥 ∈ dom (𝐹 ↾ {𝐴})) → 𝑥 = 𝐴)
58	50, 53, 57	syl2an 599	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → 𝑥 = 𝐴)
59	58	breq1d 5063	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → (𝑥(𝐹 ↾ {𝐴})𝑧 ↔ 𝐴(𝐹 ↾ {𝐴})𝑧))
60	59	biimpd 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → (𝑥(𝐹 ↾ {𝐴})𝑧 → 𝐴(𝐹 ↾ {𝐴})𝑧))
61	60	ex 416	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (𝑥(𝐹 ↾ {𝐴})𝑧 → (𝑥(𝐹 ↾ {𝐴})𝑧 → 𝐴(𝐹 ↾ {𝐴})𝑧)))
62	61	pm2.43d 53	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (𝑥(𝐹 ↾ {𝐴})𝑧 → 𝐴(𝐹 ↾ {𝐴})𝑧))
63	62	anim1d 614	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
64	63	eximdv 1925	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
65	64	exlimdv 1941	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (∃𝑥∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
66	43, 65	mpan9 510	. . . . . . . . . 10 ⊢ ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
67	9	eleq2i 2829	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ((𝐹 ↾ {𝐴}) “ {𝐴}) ↔ 𝑦 ∈ (𝐹 “ {𝐴}))
68		elimasni 5959	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ((𝐹 ↾ {𝐴}) “ {𝐴}) → 𝐴(𝐹 ↾ {𝐴})𝑦)
69	67, 68	sylbir 238	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐹 “ {𝐴}) → 𝐴(𝐹 ↾ {𝐴})𝑦)
70		vex 3412	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
71	70, 52	uniop 5398	. . . . . . . . . . . . . . . 16 ⊢ ∪ ⟨𝑦, 𝑧⟩ = {𝑦, 𝑧}
72		opex 5348	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
73	72	unisn 4841	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ {⟨𝑦, 𝑧⟩} = ⟨𝑦, 𝑧⟩
74	27	brrelex1i 5605	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴(𝐹 ↾ {𝐴})𝑧 → 𝐴 ∈ V)
75		brcnvg 5748	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ V ∧ 𝐴 ∈ V) → (𝑦◡(𝐹 ↾ {𝐴})𝐴 ↔ 𝐴(𝐹 ↾ {𝐴})𝑦))
76	70, 74, 75	sylancr 590	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴(𝐹 ↾ {𝐴})𝑧 → (𝑦◡(𝐹 ↾ {𝐴})𝐴 ↔ 𝐴(𝐹 ↾ {𝐴})𝑦))
77	76	biimpar 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → 𝑦◡(𝐹 ↾ {𝐴})𝐴)
78	74	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦◡(𝐹 ↾ {𝐴})𝐴 ∧ 𝐴(𝐹 ↾ {𝐴})𝑧) → 𝐴 ∈ V)
79		breq2 5057	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = 𝐴 → (𝑦◡(𝐹 ↾ {𝐴})𝑥 ↔ 𝑦◡(𝐹 ↾ {𝐴})𝐴))
80		breq1 5056	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = 𝐴 → (𝑥(𝐹 ↾ {𝐴})𝑧 ↔ 𝐴(𝐹 ↾ {𝐴})𝑧))
81	79, 80	anbi12d 634	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝐴 → ((𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) ↔ (𝑦◡(𝐹 ↾ {𝐴})𝐴 ∧ 𝐴(𝐹 ↾ {𝐴})𝑧)))
82	81	rspcev 3537	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ V ∧ (𝑦◡(𝐹 ↾ {𝐴})𝐴 ∧ 𝐴(𝐹 ↾ {𝐴})𝑧)) → ∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
83	78, 82	mpancom 688	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦◡(𝐹 ↾ {𝐴})𝐴 ∧ 𝐴(𝐹 ↾ {𝐴})𝑧) → ∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
84	83	ancoms 462	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ 𝑦◡(𝐹 ↾ {𝐴})𝐴) → ∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
85	77, 84	syldan 594	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
86	85	anim1i 618	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) ∧ ¬ 𝑧 = 𝑦) → (∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦))
87	86	an32s 652	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → (∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦))
88		eldif 3876	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) ↔ (⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∧ ¬ ⟨𝑦, 𝑧⟩ ∈ I ))
89		rexv 3433	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) ↔ ∃𝑥(𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
90	70, 52	brco 5739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴}))𝑧 ↔ ∃𝑥(𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
91		df-br 5054	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴}))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})))
92	89, 90, 91	3bitr2ri 303	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ↔ ∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
93	52	ideq 5721	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧)
94		df-br 5054	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 I 𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ I )
95		equcom 2026	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦)
96	93, 94, 95	3bitr3i 304	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑦, 𝑧⟩ ∈ I ↔ 𝑧 = 𝑦)
97	96	notbii 323	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ ⟨𝑦, 𝑧⟩ ∈ I ↔ ¬ 𝑧 = 𝑦)
98	92, 97	anbi12i 630	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∧ ¬ ⟨𝑦, 𝑧⟩ ∈ I ) ↔ (∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦))
99	88, 98	bitr2i 279	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦) ↔ ⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
100	87, 99	sylib 221	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
101		snssi 4721	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) → {⟨𝑦, 𝑧⟩} ⊆ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
102		uniss 4827	. . . . . . . . . . . . . . . . . . 19 ⊢ ({⟨𝑦, 𝑧⟩} ⊆ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) → ∪ {⟨𝑦, 𝑧⟩} ⊆ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
103	100, 101, 102	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ∪ {⟨𝑦, 𝑧⟩} ⊆ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
104	73, 103	eqsstrrid 3950	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ⟨𝑦, 𝑧⟩ ⊆ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
105	104	unissd 4829	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ∪ ⟨𝑦, 𝑧⟩ ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
106	71, 105	eqsstrrid 3950	. . . . . . . . . . . . . . 15 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → {𝑦, 𝑧} ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
107	70, 52	prss 4733	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) ∧ 𝑧 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) ↔ {𝑦, 𝑧} ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
108	106, 107	sylibr 237	. . . . . . . . . . . . . 14 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → (𝑦 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) ∧ 𝑧 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
109	108	simpld 498	. . . . . . . . . . . . 13 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → 𝑦 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
110	109	ex 416	. . . . . . . . . . . 12 ⊢ ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝐴(𝐹 ↾ {𝐴})𝑦 → 𝑦 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
111	69, 110	syl5 34	. . . . . . . . . . 11 ⊢ ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝑦 ∈ (𝐹 “ {𝐴}) → 𝑦 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
112	111	exlimiv 1938	. . . . . . . . . 10 ⊢ (∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝑦 ∈ (𝐹 “ {𝐴}) → 𝑦 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
113	66, 112	syl 17	. . . . . . . . 9 ⊢ ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → (𝑦 ∈ (𝐹 “ {𝐴}) → 𝑦 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
114	113	ssrdv 3907	. . . . . . . 8 ⊢ ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → (𝐹 “ {𝐴}) ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
115		ssdif0 4278	. . . . . . . 8 ⊢ ((𝐹 “ {𝐴}) ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) ↔ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
116	114, 115	sylib 221	. . . . . . 7 ⊢ ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
117	116	ex 416	. . . . . 6 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅))
118		ndmima 5971	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ∅)
119	9, 118	eqtr3id 2792	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (𝐹 “ {𝐴}) = ∅)
120	119	difeq1d 4036	. . . . . . 7 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = (∅ ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
121		0dif 4316	. . . . . . 7 ⊢ (∅ ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅
122	120, 121	eqtrdi 2794	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
123	117, 122	pm2.61d1 183	. . . . 5 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
124	123	unieqd 4833	. . . 4 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∪ ∅)
125	124, 17	eqtrdi 2794	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
126	26, 125	eqtr4d 2780	. 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
127	25, 126	pm2.61i 185	1 ⊢ (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541   = wceq 1543  ∃wex 1787   ∈ wcel 2110  ∃wrex 3062  Vcvv 3408   ∖ cdif 3863   ⊆ wss 3866  ∅c0 4237  {csn 4541  {cpr 4543  ⟨cop 4547  ∪ cuni 4819   class class class wbr 5053   I cid 5454  ^◡ccnv 5550  dom cdm 5551   ↾ cres 5553   “ cima 5554   ∘ ccom 5555  Rel wrel 5556  Fun wfun 6374  ‘cfv 6380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388

This theorem is referenced by: (None)