Theorem dffv2 6930

Description: Alternate definition of function value df-fv 6501 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 4619	. . . . 5 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
2		fvres 6854	. . . . 5 ⊢ (𝐴 ∈ {𝐴} → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴))
3	1, 2	sylbi 217	. . . 4 ⊢ (𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴))
4		fvprc 6827	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = ∅)
5		fvprc 6827	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
6	4, 5	eqtr4d 2775	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴))
7	3, 6	pm2.61i 182	. . 3 ⊢ ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴)
8		funfv 6922	. . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴})‘𝐴) = ∪ ((𝐹 ↾ {𝐴}) “ {𝐴}))
9		resima 5975	. . . . . . 7 ⊢ ((𝐹 ↾ {𝐴}) “ {𝐴}) = (𝐹 “ {𝐴})
10		dif0 4331	. . . . . . 7 ⊢ ((𝐹 “ {𝐴}) ∖ ∅) = (𝐹 “ {𝐴})
11	9, 10	eqtr4i 2763	. . . . . 6 ⊢ ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ ∅)
12		df-fun 6495	. . . . . . . . . . . . 13 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ⊆ I ))
13	12	simprbi 496	. . . . . . . . . . . 12 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ⊆ I )
14		ssdif0 4319	. . . . . . . . . . . 12 ⊢ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ⊆ I ↔ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∅)
15	13, 14	sylib 218	. . . . . . . . . . 11 ⊢ (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∅)
16	15	unieqd 4877	. . . . . . . . . 10 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∪ ∅)
17		uni0 4892	. . . . . . . . . 10 ⊢ ∪ ∅ = ∅
18	16, 17	eqtrdi 2788	. . . . . . . . 9 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∅)
19	18	unieqd 4877	. . . . . . . 8 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∪ ∅)
20	19, 17	eqtrdi 2788	. . . . . . 7 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∅)
21	20	difeq2d 4079	. . . . . 6 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ((𝐹 “ {𝐴}) ∖ ∅))
22	11, 21	eqtr4id 2791	. . . . 5 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
23	22	unieqd 4877	. . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ ((𝐹 ↾ {𝐴}) “ {𝐴}) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
24	8, 23	eqtrd 2772	. . 3 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴})‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
25	7, 24	eqtr3id 2786	. 2 ⊢ (Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
26		nfunsn 6874	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)
27		relres 5965	. . . . . . . . . . . . . . 15 ⊢ Rel (𝐹 ↾ {𝐴})
28		dffun3 6505	. . . . . . . . . . . . . . 15 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦)))
29	27, 28	mpbiran 710	. . . . . . . . . . . . . 14 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ ∀𝑥∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦))
30		iman 401	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
31	30	albii 1821	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧 ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
32		alnex 1783	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑧 ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
33	31, 32	bitri 275	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
34	33	exbii 1850	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ∃𝑦 ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
35		exnal 1829	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑦 ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
36	34, 35	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ¬ ∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
37	36	albii 1821	. . . . . . . . . . . . . 14 ⊢ (∀𝑥∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ∀𝑥 ¬ ∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
38		alnex 1783	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ¬ ∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∃𝑥∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
39	29, 37, 38	3bitrri 298	. . . . . . . . . . . . 13 ⊢ (¬ ∃𝑥∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ Fun (𝐹 ↾ {𝐴}))
40	39	con1bii 356	. . . . . . . . . . . 12 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) ↔ ∃𝑥∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
41		sp 2191	. . . . . . . . . . . . 13 ⊢ (∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
42	41	eximi 1837	. . . . . . . . . . . 12 ⊢ (∃𝑥∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑥∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
43	40, 42	sylbi 217	. . . . . . . . . . 11 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ∃𝑥∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
44		snssi 4765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → {𝐴} ⊆ dom (𝐹 ↾ {𝐴}))
45		residm 5970	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = (𝐹 ↾ {𝐴})
46	45	dmeqi 5854	. . . . . . . . . . . . . . . . . . . . 21 ⊢ dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = dom (𝐹 ↾ {𝐴})
47		ssdmres 5973	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) ↔ dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = {𝐴})
48	47	biimpi 216	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) → dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = {𝐴})
49	46, 48	eqtr3id 2786	. . . . . . . . . . . . . . . . . . . 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 3445	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
52		vex 3445	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑧 ∈ V
53	51, 52	breldm 5858	. . . . . . . . . . . . . . . . . . 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 4597	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
56	54, 55	bitrdi 287	. . . . . . . . . . . . . . . . . . . 20 ⊢ (dom (𝐹 ↾ {𝐴}) = {𝐴} → (𝑥 ∈ dom (𝐹 ↾ {𝐴}) ↔ 𝑥 = 𝐴))
57	56	biimpa 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom (𝐹 ↾ {𝐴}) = {𝐴} ∧ 𝑥 ∈ dom (𝐹 ↾ {𝐴})) → 𝑥 = 𝐴)
58	50, 53, 57	syl2an 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → 𝑥 = 𝐴)
59	58	breq1d 5109	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → (𝑥(𝐹 ↾ {𝐴})𝑧 ↔ 𝐴(𝐹 ↾ {𝐴})𝑧))
60	59	biimpd 229	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → (𝑥(𝐹 ↾ {𝐴})𝑧 → 𝐴(𝐹 ↾ {𝐴})𝑧))
61	60	ex 412	. . . . . . . . . . . . . . 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 612	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
64	63	eximdv 1919	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
65	64	exlimdv 1935	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (∃𝑥∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
66	43, 65	mpan9 506	. . . . . . . . . 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 6051	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ((𝐹 ↾ {𝐴}) “ {𝐴}) → 𝐴(𝐹 ↾ {𝐴})𝑦)
69	67, 68	sylbir 235	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐹 “ {𝐴}) → 𝐴(𝐹 ↾ {𝐴})𝑦)
70		vex 3445	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
71	70, 52	uniop 5464	. . . . . . . . . . . . . . . 16 ⊢ ∪ ⟨𝑦, 𝑧⟩ = {𝑦, 𝑧}
72		opex 5413	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
73	72	unisn 4883	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ {⟨𝑦, 𝑧⟩} = ⟨𝑦, 𝑧⟩
74	27	brrelex1i 5681	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴(𝐹 ↾ {𝐴})𝑧 → 𝐴 ∈ V)
75		brcnvg 5829	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ V ∧ 𝐴 ∈ V) → (𝑦◡(𝐹 ↾ {𝐴})𝐴 ↔ 𝐴(𝐹 ↾ {𝐴})𝑦))
76	70, 74, 75	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴(𝐹 ↾ {𝐴})𝑧 → (𝑦◡(𝐹 ↾ {𝐴})𝐴 ↔ 𝐴(𝐹 ↾ {𝐴})𝑦))
77	76	biimpar 477	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → 𝑦◡(𝐹 ↾ {𝐴})𝐴)
78	74	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦◡(𝐹 ↾ {𝐴})𝐴 ∧ 𝐴(𝐹 ↾ {𝐴})𝑧) → 𝐴 ∈ V)
79		breq2 5103	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = 𝐴 → (𝑦◡(𝐹 ↾ {𝐴})𝑥 ↔ 𝑦◡(𝐹 ↾ {𝐴})𝐴))
80		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = 𝐴 → (𝑥(𝐹 ↾ {𝐴})𝑧 ↔ 𝐴(𝐹 ↾ {𝐴})𝑧))
81	79, 80	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝐴 → ((𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) ↔ (𝑦◡(𝐹 ↾ {𝐴})𝐴 ∧ 𝐴(𝐹 ↾ {𝐴})𝑧)))
82	81	rspcev 3577	. . . . . . . . . . . . . . . . . . . . . . . . 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 689	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦◡(𝐹 ↾ {𝐴})𝐴 ∧ 𝐴(𝐹 ↾ {𝐴})𝑧) → ∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
84	83	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ 𝑦◡(𝐹 ↾ {𝐴})𝐴) → ∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
85	77, 84	syldan 592	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
86	85	anim1i 616	. . . . . . . . . . . . . . . . . . . . 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 653	. . . . . . . . . . . . . . . . . . . 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 3912	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) ↔ (⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∧ ¬ ⟨𝑦, 𝑧⟩ ∈ I ))
89		rexv 3469	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) ↔ ∃𝑥(𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
90	70, 52	brco 5820	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴}))𝑧 ↔ ∃𝑥(𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
91		df-br 5100	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴}))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})))
92	89, 90, 91	3bitr2ri 300	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ↔ ∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
93	52	ideq 5802	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧)
94		df-br 5100	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 I 𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ I )
95		equcom 2020	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦)
96	93, 94, 95	3bitr3i 301	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑦, 𝑧⟩ ∈ I ↔ 𝑧 = 𝑦)
97	96	notbii 320	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ ⟨𝑦, 𝑧⟩ ∈ I ↔ ¬ 𝑧 = 𝑦)
98	92, 97	anbi12i 629	. . . . . . . . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦) ↔ ⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
100	87, 99	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
101		snssi 4765	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) → {⟨𝑦, 𝑧⟩} ⊆ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
102		uniss 4872	. . . . . . . . . . . . . . . . . . 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 3974	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ⟨𝑦, 𝑧⟩ ⊆ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
105	104	unissd 4874	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ∪ ⟨𝑦, 𝑧⟩ ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
106	71, 105	eqsstrrid 3974	. . . . . . . . . . . . . . 15 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → {𝑦, 𝑧} ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
107	70, 52	prss 4777	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) ∧ 𝑧 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) ↔ {𝑦, 𝑧} ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
108	106, 107	sylibr 234	. . . . . . . . . . . . . 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 494	. . . . . . . . . . . . 13 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → 𝑦 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
110	109	ex 412	. . . . . . . . . . . 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 1932	. . . . . . . . . 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 3940	. . . . . . . 8 ⊢ ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → (𝐹 “ {𝐴}) ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
115		ssdif0 4319	. . . . . . . 8 ⊢ ((𝐹 “ {𝐴}) ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) ↔ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
116	114, 115	sylib 218	. . . . . . 7 ⊢ ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
117	116	ex 412	. . . . . 6 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅))
118		ndmima 6063	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ∅)
119	9, 118	eqtr3id 2786	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (𝐹 “ {𝐴}) = ∅)
120	119	difeq1d 4078	. . . . . . 7 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = (∅ ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
121		0dif 4358	. . . . . . 7 ⊢ (∅ ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅
122	120, 121	eqtrdi 2788	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
123	117, 122	pm2.61d1 180	. . . . 5 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
124	123	unieqd 4877	. . . 4 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∪ ∅)
125	124, 17	eqtrdi 2788	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
126	26, 125	eqtr4d 2775	. 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
127	25, 126	pm2.61i 182	1 ⊢ (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3061  Vcvv 3441   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4286  {csn 4581  {cpr 4583  ⟨cop 4587  ∪ cuni 4864   class class class wbr 5099   I cid 5519  ^◡ccnv 5624  dom cdm 5625   ↾ cres 5627   “ cima 5628   ∘ ccom 5629  Rel wrel 5630  Fun wfun 6487  ‘cfv 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501

This theorem is referenced by: (None)