Theorem tgjustf 26834

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		relopabv 5731	. . 3 ⊢ Rel {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}
2		ancom 461	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
3		eqcom 2745	. . . . . 6 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝑥))
4	2, 3	anbi12i 627	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑥)))
5		simpl 483	. . . . . . . 8 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → 𝑢 = 𝑥)
6	5	fveq2d 6778	. . . . . . 7 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → (𝐹‘𝑢) = (𝐹‘𝑥))
7		simpr 485	. . . . . . . 8 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → 𝑣 = 𝑦)
8	7	fveq2d 6778	. . . . . . 7 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → (𝐹‘𝑣) = (𝐹‘𝑦))
9	6, 8	eqeq12d 2754	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
10		eqid 2738	. . . . . 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 5680	. . . . 5 ⊢ (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)))
12		simpl 483	. . . . . . . 8 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → 𝑢 = 𝑦)
13	12	fveq2d 6778	. . . . . . 7 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → (𝐹‘𝑢) = (𝐹‘𝑦))
14		simpr 485	. . . . . . . 8 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → 𝑣 = 𝑥)
15	14	fveq2d 6778	. . . . . . 7 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → (𝐹‘𝑣) = (𝐹‘𝑥))
16	13, 15	eqeq12d 2754	. . . . . 6 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑦) = (𝐹‘𝑥)))
17	16, 10	brab2a 5680	. . . . 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 303	. . . 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 215	. . 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 772	. . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → 𝑥 ∈ 𝐴)
21		simprlr 777	. . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → 𝑧 ∈ 𝐴)
22		simplr 766	. . . . . 6 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → (𝐹‘𝑥) = (𝐹‘𝑦))
23		simprr 770	. . . . . 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 2778	. . . . 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 515	. . . 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 483	. . . . . . . 8 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → 𝑢 = 𝑦)
27	26	fveq2d 6778	. . . . . . 7 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → (𝐹‘𝑢) = (𝐹‘𝑦))
28		simpr 485	. . . . . . . 8 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → 𝑣 = 𝑧)
29	28	fveq2d 6778	. . . . . . 7 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → (𝐹‘𝑣) = (𝐹‘𝑧))
30	27, 29	eqeq12d 2754	. . . . . 6 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑦) = (𝐹‘𝑧)))
31	30, 10	brab2a 5680	. . . . 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 627	. . . 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 483	. . . . . . 7 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → 𝑢 = 𝑥)
34	33	fveq2d 6778	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → (𝐹‘𝑢) = (𝐹‘𝑥))
35		simpr 485	. . . . . . 7 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → 𝑣 = 𝑧)
36	35	fveq2d 6778	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → (𝐹‘𝑣) = (𝐹‘𝑧))
37	34, 36	eqeq12d 2754	. . . . 5 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑥) = (𝐹‘𝑧)))
38	37, 10	brab2a 5680	. . . 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 292	. . 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 2738	. . . . 5 ⊢ (𝐹‘𝑥) = (𝐹‘𝑥)
41	40	biantru 530	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑥)))
42		pm4.24 564	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
43		simpl 483	. . . . . . 7 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → 𝑢 = 𝑥)
44	43	fveq2d 6778	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → (𝐹‘𝑢) = (𝐹‘𝑥))
45		simpr 485	. . . . . . 7 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → 𝑣 = 𝑥)
46	45	fveq2d 6778	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → (𝐹‘𝑣) = (𝐹‘𝑥))
47	44, 46	eqeq12d 2754	. . . . 5 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑥) = (𝐹‘𝑥)))
48	47, 10	brab2a 5680	. . . 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 303	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑥)
50	1, 19, 39, 49	iseri 8525	. 2 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} Er 𝐴
51	11	baib 536	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
52	51	rgen2 3120	. 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 776	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))) → 𝑢 ∈ 𝐴)
55		simprlr 777	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))) → 𝑣 ∈ 𝐴)
56	53, 53, 54, 55	opabex2 7897	. . 3 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} ∈ V)
57		ereq1 8505	. . . . 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 483	. . . . . . . 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 5085	. . . . . . 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 344	. . . . . 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 3128	. . . . 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 631	. . . 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 3536	. . 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 695	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   class class class wbr 5074  {copab 5136  ‘cfv 6433   Er wer 8495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fv 6441  df-er 8498

This theorem is referenced by:  tgjustc1  26836