Theorem dffv2 6929

Description: Alternate definition of function value df-fv 6500 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 4600	. . . . 5 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
2		fvres 6853	. . . . 5 ⊢ (𝐴 ∈ {𝐴} → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴))
3	1, 2	sylbi 218	. . . 4 ⊢ (𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴))
4		fvprc 6826	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = ∅)
5		fvprc 6826	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
6	4, 5	eqtr4d 2778	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴))
7	3, 6	pm2.61i 183	. . 3 ⊢ ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹‘𝐴)
8		funfv 6921	. . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴})‘𝐴) = ∪ ((𝐹 ↾ {𝐴}) “ {𝐴}))
9		resima 5974	. . . . . . 7 ⊢ ((𝐹 ↾ {𝐴}) “ {𝐴}) = (𝐹 “ {𝐴})
10		dif0 4313	. . . . . . 7 ⊢ ((𝐹 “ {𝐴}) ∖ ∅) = (𝐹 “ {𝐴})
11	9, 10	eqtr4i 2766	. . . . . 6 ⊢ ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ ∅)
12		df-fun 6494	. . . . . . . . . . . . 13 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ⊆ I ))
13	12	simprbi 498	. . . . . . . . . . . 12 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ⊆ I )
14		ssdif0 4301	. . . . . . . . . . . 12 ⊢ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ⊆ I ↔ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∅)
15	13, 14	sylib 219	. . . . . . . . . . 11 ⊢ (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∅)
16	15	unieqd 4858	. . . . . . . . . 10 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∪ ∅)
17		uni0 4873	. . . . . . . . . 10 ⊢ ∪ ∅ = ∅
18	16, 17	eqtrdi 2791	. . . . . . . . 9 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∅)
19	18	unieqd 4858	. . . . . . . 8 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∪ ∅)
20	19, 17	eqtrdi 2791	. . . . . . 7 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) = ∅)
21	20	difeq2d 4064	. . . . . 6 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ((𝐹 “ {𝐴}) ∖ ∅))
22	11, 21	eqtr4id 2794	. . . . 5 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
23	22	unieqd 4858	. . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) → ∪ ((𝐹 ↾ {𝐴}) “ {𝐴}) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
24	8, 23	eqtrd 2775	. . 3 ⊢ (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴})‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
25	7, 24	eqtr3id 2789	. 2 ⊢ (Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
26		nfunsn 6873	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)
27		relres 5964	. . . . . . . . . . . . . . 15 ⊢ Rel (𝐹 ↾ {𝐴})
28		dffun3 6504	. . . . . . . . . . . . . . 15 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦)))
29	27, 28	mpbiran 715	. . . . . . . . . . . . . 14 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ ∀𝑥∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦))
30		iman 402	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
31	30	albii 1826	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧 ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
32		alnex 1788	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑧 ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
33	31, 32	bitri 276	. . . . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ¬ ∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
37	36	albii 1826	. . . . . . . . . . . . . 14 ⊢ (∀𝑥∃𝑦∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑧 = 𝑦) ↔ ∀𝑥 ¬ ∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
38		alnex 1788	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ¬ ∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∃𝑥∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
39	29, 37, 38	3bitrri 299	. . . . . . . . . . . . 13 ⊢ (¬ ∃𝑥∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ Fun (𝐹 ↾ {𝐴}))
40	39	con1bii 357	. . . . . . . . . . . 12 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) ↔ ∃𝑥∀𝑦∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
41		sp 2195	. . . . . . . . . . . . 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 218	. . . . . . . . . . 11 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ∃𝑥∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
44		snssi 4724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → {𝐴} ⊆ dom (𝐹 ↾ {𝐴}))
45		residm 5969	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = (𝐹 ↾ {𝐴})
46	45	dmeqi 5853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = dom (𝐹 ↾ {𝐴})
47		ssdmres 5972	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) ↔ dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = {𝐴})
48	47	biimpi 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) → dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = {𝐴})
49	46, 48	eqtr3id 2789	. . . . . . . . . . . . . . . . . . . 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 3436	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
52		vex 3436	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑧 ∈ V
53	51, 52	breldm 5857	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑧 → 𝑥 ∈ dom (𝐹 ↾ {𝐴}))
54		eleq2 2829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (dom (𝐹 ↾ {𝐴}) = {𝐴} → (𝑥 ∈ dom (𝐹 ↾ {𝐴}) ↔ 𝑥 ∈ {𝐴}))
55		velsn 4578	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
56	54, 55	bitrdi 288	. . . . . . . . . . . . . . . . . . . 20 ⊢ (dom (𝐹 ↾ {𝐴}) = {𝐴} → (𝑥 ∈ dom (𝐹 ↾ {𝐴}) ↔ 𝑥 = 𝐴))
57	56	biimpa 477	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom (𝐹 ↾ {𝐴}) = {𝐴} ∧ 𝑥 ∈ dom (𝐹 ↾ {𝐴})) → 𝑥 = 𝐴)
58	50, 53, 57	syl2an 602	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → 𝑥 = 𝐴)
59	58	breq1d 5089	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → (𝑥(𝐹 ↾ {𝐴})𝑧 ↔ 𝐴(𝐹 ↾ {𝐴})𝑧))
60	59	biimpd 230	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → (𝑥(𝐹 ↾ {𝐴})𝑧 → 𝐴(𝐹 ↾ {𝐴})𝑧))
61	60	ex 413	. . . . . . . . . . . . . . 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 617	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
64	63	eximdv 1924	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
65	64	exlimdv 1940	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (∃𝑥∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
66	43, 65	mpan9 511	. . . . . . . . . 10 ⊢ ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
67	9	eleq2i 2832	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ((𝐹 ↾ {𝐴}) “ {𝐴}) ↔ 𝑦 ∈ (𝐹 “ {𝐴}))
68		elimasni 6050	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ((𝐹 ↾ {𝐴}) “ {𝐴}) → 𝐴(𝐹 ↾ {𝐴})𝑦)
69	67, 68	sylbir 236	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐹 “ {𝐴}) → 𝐴(𝐹 ↾ {𝐴})𝑦)
70		vex 3436	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
71	70, 52	uniop 5463	. . . . . . . . . . . . . . . 16 ⊢ ∪ ⟨𝑦, 𝑧⟩ = {𝑦, 𝑧}
72		opex 5410	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
73	72	unisn 4864	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ {⟨𝑦, 𝑧⟩} = ⟨𝑦, 𝑧⟩
74	27	brrelex1i 5681	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴(𝐹 ↾ {𝐴})𝑧 → 𝐴 ∈ V)
75		brcnvg 5828	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ V ∧ 𝐴 ∈ V) → (𝑦◡(𝐹 ↾ {𝐴})𝐴 ↔ 𝐴(𝐹 ↾ {𝐴})𝑦))
76	70, 74, 75	sylancr 593	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴(𝐹 ↾ {𝐴})𝑧 → (𝑦◡(𝐹 ↾ {𝐴})𝐴 ↔ 𝐴(𝐹 ↾ {𝐴})𝑦))
77	76	biimpar 478	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → 𝑦◡(𝐹 ↾ {𝐴})𝐴)
78	74	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦◡(𝐹 ↾ {𝐴})𝐴 ∧ 𝐴(𝐹 ↾ {𝐴})𝑧) → 𝐴 ∈ V)
79		breq2 5083	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = 𝐴 → (𝑦◡(𝐹 ↾ {𝐴})𝑥 ↔ 𝑦◡(𝐹 ↾ {𝐴})𝐴))
80		breq1 5082	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = 𝐴 → (𝑥(𝐹 ↾ {𝐴})𝑧 ↔ 𝐴(𝐹 ↾ {𝐴})𝑧))
81	79, 80	anbi12d 638	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝐴 → ((𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) ↔ (𝑦◡(𝐹 ↾ {𝐴})𝐴 ∧ 𝐴(𝐹 ↾ {𝐴})𝑧)))
82	81	rspcev 3567	. . . . . . . . . . . . . . . . . . . . . . . . 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 694	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦◡(𝐹 ↾ {𝐴})𝐴 ∧ 𝐴(𝐹 ↾ {𝐴})𝑧) → ∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
84	83	ancoms 459	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ 𝑦◡(𝐹 ↾ {𝐴})𝐴) → ∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
85	77, 84	syldan 597	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
86	85	anim1i 621	. . . . . . . . . . . . . . . . . . . . 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 658	. . . . . . . . . . . . . . . . . . . 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 3900	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) ↔ (⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∧ ¬ ⟨𝑦, 𝑧⟩ ∈ I ))
89		rexv 3460	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) ↔ ∃𝑥(𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
90	70, 52	brco 5819	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴}))𝑧 ↔ ∃𝑥(𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
91		df-br 5080	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴}))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})))
92	89, 90, 91	3bitr2ri 301	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ↔ ∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧))
93	52	ideq 5801	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧)
94		df-br 5080	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 I 𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ I )
95		equcom 2025	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦)
96	93, 94, 95	3bitr3i 302	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑦, 𝑧⟩ ∈ I ↔ 𝑧 = 𝑦)
97	96	notbii 321	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ ⟨𝑦, 𝑧⟩ ∈ I ↔ ¬ 𝑧 = 𝑦)
98	92, 97	anbi12i 634	. . . . . . . . . . . . . . . . . . . . 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 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃𝑥 ∈ V (𝑦◡(𝐹 ↾ {𝐴})𝑥 ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦) ↔ ⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
100	87, 99	sylib 219	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
101		snssi 4724	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) → {⟨𝑦, 𝑧⟩} ⊆ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
102		uniss 4853	. . . . . . . . . . . . . . . . . . 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 3961	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ⟨𝑦, 𝑧⟩ ⊆ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
105	104	unissd 4855	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ∪ ⟨𝑦, 𝑧⟩ ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
106	71, 105	eqsstrrid 3961	. . . . . . . . . . . . . . 15 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → {𝑦, 𝑧} ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
107	70, 52	prss 4758	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) ∧ 𝑧 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) ↔ {𝑦, 𝑧} ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
108	106, 107	sylibr 235	. . . . . . . . . . . . . 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 495	. . . . . . . . . . . . 13 ⊢ (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → 𝑦 ∈ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
110	109	ex 413	. . . . . . . . . . . 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 1937	. . . . . . . . . 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 3928	. . . . . . . 8 ⊢ ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → (𝐹 “ {𝐴}) ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
115		ssdif0 4301	. . . . . . . 8 ⊢ ((𝐹 “ {𝐴}) ⊆ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ) ↔ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
116	114, 115	sylib 219	. . . . . . 7 ⊢ ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
117	116	ex 413	. . . . . 6 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅))
118		ndmima 6062	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ∅)
119	9, 118	eqtr3id 2789	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (𝐹 “ {𝐴}) = ∅)
120	119	difeq1d 4063	. . . . . . 7 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = (∅ ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
121		0dif 4340	. . . . . . 7 ⊢ (∅ ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅
122	120, 121	eqtrdi 2791	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
123	117, 122	pm2.61d1 181	. . . . 5 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
124	123	unieqd 4858	. . . 4 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∪ ∅)
125	124, 17	eqtrdi 2791	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )) = ∅)
126	26, 125	eqtr4d 2778	. 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I )))
127	25, 126	pm2.61i 183	1 ⊢ (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃wrex 3064  Vcvv 3432   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4268  {csn 4562  {cpr 4564  ⟨cop 4568  ∪ cuni 4845   class class class wbr 5079   I cid 5519  ^◡ccnv 5624  dom cdm 5625   ↾ cres 5627   “ cima 5628   ∘ ccom 5629  Rel wrel 5630  Fun wfun 6486  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  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 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by: (None)