Theorem tgjustf 26267

Description: Given any function 𝐹, equality of the image by 𝐹 is an equivalence relation. (Contributed by Thierry Arnoux, 25-Jan-2023.)

Assertion

Ref	Expression
tgjustf	⊢ (𝐴 ∈ 𝑉 → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))))

Distinct variable groups:   𝐴,𝑟,𝑥,𝑦   𝐹,𝑟,𝑥,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑟)

Proof of Theorem tgjustf

Dummy variables 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 5660	. . 3 ⊢ Rel {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}
2		ancom 464	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
3		eqcom 2805	. . . . . 6 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝑥))
4	2, 3	anbi12i 629	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑥)))
5		simpl 486	. . . . . . . 8 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → 𝑢 = 𝑥)
6	5	fveq2d 6649	. . . . . . 7 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → (𝐹‘𝑢) = (𝐹‘𝑥))
7		simpr 488	. . . . . . . 8 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → 𝑣 = 𝑦)
8	7	fveq2d 6649	. . . . . . 7 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → (𝐹‘𝑣) = (𝐹‘𝑦))
9	6, 8	eqeq12d 2814	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
10		eqid 2798	. . . . . 6 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}
11	9, 10	brab2a 5608	. . . . 5 ⊢ (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)))
12		simpl 486	. . . . . . . 8 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → 𝑢 = 𝑦)
13	12	fveq2d 6649	. . . . . . 7 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → (𝐹‘𝑢) = (𝐹‘𝑦))
14		simpr 488	. . . . . . . 8 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → 𝑣 = 𝑥)
15	14	fveq2d 6649	. . . . . . 7 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → (𝐹‘𝑣) = (𝐹‘𝑥))
16	13, 15	eqeq12d 2814	. . . . . 6 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑦) = (𝐹‘𝑥)))
17	16, 10	brab2a 5608	. . . . 5 ⊢ (𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑥 ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑥)))
18	4, 11, 17	3bitr4i 306	. . . 4 ⊢ (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ 𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑥)
19	18	biimpi 219	. . 3 ⊢ (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 → 𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑥)
20		simplll 774	. . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → 𝑥 ∈ 𝐴)
21		simprlr 779	. . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → 𝑧 ∈ 𝐴)
22		simplr 768	. . . . . 6 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → (𝐹‘𝑥) = (𝐹‘𝑦))
23		simprr 772	. . . . . 6 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → (𝐹‘𝑦) = (𝐹‘𝑧))
24	22, 23	eqtrd 2833	. . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → (𝐹‘𝑥) = (𝐹‘𝑧))
25	20, 21, 24	jca31 518	. . . 4 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑧)))
26		simpl 486	. . . . . . . 8 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → 𝑢 = 𝑦)
27	26	fveq2d 6649	. . . . . . 7 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → (𝐹‘𝑢) = (𝐹‘𝑦))
28		simpr 488	. . . . . . . 8 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → 𝑣 = 𝑧)
29	28	fveq2d 6649	. . . . . . 7 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → (𝐹‘𝑣) = (𝐹‘𝑧))
30	27, 29	eqeq12d 2814	. . . . . 6 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑦) = (𝐹‘𝑧)))
31	30, 10	brab2a 5608	. . . . 5 ⊢ (𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑧 ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)))
32	11, 31	anbi12i 629	. . . 4 ⊢ ((𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ∧ 𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑧) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))))
33		simpl 486	. . . . . . 7 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → 𝑢 = 𝑥)
34	33	fveq2d 6649	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → (𝐹‘𝑢) = (𝐹‘𝑥))
35		simpr 488	. . . . . . 7 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → 𝑣 = 𝑧)
36	35	fveq2d 6649	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → (𝐹‘𝑣) = (𝐹‘𝑧))
37	34, 36	eqeq12d 2814	. . . . 5 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑥) = (𝐹‘𝑧)))
38	37, 10	brab2a 5608	. . . 4 ⊢ (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑧)))
39	25, 32, 38	3imtr4i 295	. . 3 ⊢ ((𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ∧ 𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑧) → 𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑧)
40		eqid 2798	. . . . 5 ⊢ (𝐹‘𝑥) = (𝐹‘𝑥)
41	40	biantru 533	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑥)))
42		pm4.24 567	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
43		simpl 486	. . . . . . 7 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → 𝑢 = 𝑥)
44	43	fveq2d 6649	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → (𝐹‘𝑢) = (𝐹‘𝑥))
45		simpr 488	. . . . . . 7 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → 𝑣 = 𝑥)
46	45	fveq2d 6649	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → (𝐹‘𝑣) = (𝐹‘𝑥))
47	44, 46	eqeq12d 2814	. . . . 5 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑥) = (𝐹‘𝑥)))
48	47, 10	brab2a 5608	. . . 4 ⊢ (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑥 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑥)))
49	41, 42, 48	3bitr4i 306	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑥)
50	1, 19, 39, 49	iseri 8299	. 2 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} Er 𝐴
51	11	baib 539	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
52	51	rgen2 3168	. 2 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))
53		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
54		simprll 778	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))) → 𝑢 ∈ 𝐴)
55		simprlr 779	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))) → 𝑣 ∈ 𝐴)
56	53, 53, 54, 55	opabex2 7737	. . 3 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} ∈ V)
57		ereq1 8279	. . . . 5 ⊢ (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} → (𝑟 Er 𝐴 ↔ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} Er 𝐴))
58		simpl 486	. . . . . . . 8 ⊢ ((𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))})
59	58	breqd 5041	. . . . . . 7 ⊢ ((𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑟𝑦 ↔ 𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦))
60	59	bibi1d 347	. . . . . 6 ⊢ ((𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))))
61	60	2ralbidva 3163	. . . . 5 ⊢ (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))))
62	57, 61	anbi12d 633	. . . 4 ⊢ (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} → ((𝑟 Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))) ↔ ({⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))))
63	62	spcegv 3545	. . 3 ⊢ ({⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} ∈ V → (({⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))) → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))))
64	56, 63	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (({⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))) → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))))
65	50, 52, 64	mp2ani 697	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   class class class wbr 5030  {copab 5092  ‘cfv 6324   Er wer 8269

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fv 6332  df-er 8272

This theorem is referenced by:  tgjustc1  26269