Theorem f1o2ndf1 5791

Description: The 2^nd (second member of an ordered pair) function restricted to a one-to-one function 𝐹 is a one-to-one function of 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)

Assertion

Ref	Expression
f1o2ndf1	⊢ (𝐹:A–1-1→B → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹)

Proof of Theorem f1o2ndf1

Dummy variables 𝑎 𝑏 v w x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1f 5035	. . 3 ⊢ (𝐹:A–1-1→B → 𝐹:A⟶B)
2		fo2ndf 5790	. . 3 ⊢ (𝐹:A⟶B → (2nd ↾ 𝐹):𝐹–onto→ran 𝐹)
3	1, 2	syl 14	. 2 ⊢ (𝐹:A–1-1→B → (2nd ↾ 𝐹):𝐹–onto→ran 𝐹)
4		f2ndf 5789	. . . . 5 ⊢ (𝐹:A⟶B → (2nd ↾ 𝐹):𝐹⟶B)
5	1, 4	syl 14	. . . 4 ⊢ (𝐹:A–1-1→B → (2nd ↾ 𝐹):𝐹⟶B)
6		fssxp 5001	. . . . . . 7 ⊢ (𝐹:A⟶B → 𝐹 ⊆ (A × B))
7	1, 6	syl 14	. . . . . 6 ⊢ (𝐹:A–1-1→B → 𝐹 ⊆ (A × B))
8		ssel2 2934	. . . . . . . . . . 11 ⊢ ((𝐹 ⊆ (A × B) ∧ x ∈ 𝐹) → x ∈ (A × B))
9		elxp2 4306	. . . . . . . . . . 11 ⊢ (x ∈ (A × B) ↔ ∃𝑎 ∈ A ∃v ∈ B x = ⟨𝑎, v⟩)
10	8, 9	sylib 127	. . . . . . . . . 10 ⊢ ((𝐹 ⊆ (A × B) ∧ x ∈ 𝐹) → ∃𝑎 ∈ A ∃v ∈ B x = ⟨𝑎, v⟩)
11		ssel2 2934	. . . . . . . . . . 11 ⊢ ((𝐹 ⊆ (A × B) ∧ y ∈ 𝐹) → y ∈ (A × B))
12		elxp2 4306	. . . . . . . . . . 11 ⊢ (y ∈ (A × B) ↔ ∃𝑏 ∈ A ∃w ∈ B y = ⟨𝑏, w⟩)
13	11, 12	sylib 127	. . . . . . . . . 10 ⊢ ((𝐹 ⊆ (A × B) ∧ y ∈ 𝐹) → ∃𝑏 ∈ A ∃w ∈ B y = ⟨𝑏, w⟩)
14	10, 13	anim12dan 532	. . . . . . . . 9 ⊢ ((𝐹 ⊆ (A × B) ∧ (x ∈ 𝐹 ∧ y ∈ 𝐹)) → (∃𝑎 ∈ A ∃v ∈ B x = ⟨𝑎, v⟩ ∧ ∃𝑏 ∈ A ∃w ∈ B y = ⟨𝑏, w⟩))
15		fvres 5141	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨𝑎, v⟩ ∈ 𝐹 → ((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = (2nd ‘⟨𝑎, v⟩))
16	15	adantr 261	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) → ((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = (2nd ‘⟨𝑎, v⟩))
17	16	adantr 261	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) ∧ ((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B))) → ((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = (2nd ‘⟨𝑎, v⟩))
18		fvres 5141	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨𝑏, w⟩ ∈ 𝐹 → ((2nd ↾ 𝐹)‘⟨𝑏, w⟩) = (2nd ‘⟨𝑏, w⟩))
19	18	ad2antlr 458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) ∧ ((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B))) → ((2nd ↾ 𝐹)‘⟨𝑏, w⟩) = (2nd ‘⟨𝑏, w⟩))
20	17, 19	eqeq12d 2051	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) ∧ ((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B))) → (((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = ((2nd ↾ 𝐹)‘⟨𝑏, w⟩) ↔ (2nd ‘⟨𝑎, v⟩) = (2nd ‘⟨𝑏, w⟩)))
21		vex 2554	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑎 ∈ V
22		vex 2554	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ v ∈ V
23	21, 22	op2nd 5716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2nd ‘⟨𝑎, v⟩) = v
24		vex 2554	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑏 ∈ V
25		vex 2554	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ w ∈ V
26	24, 25	op2nd 5716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2nd ‘⟨𝑏, w⟩) = w
27	23, 26	eqeq12i 2050	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((2nd ‘⟨𝑎, v⟩) = (2nd ‘⟨𝑏, w⟩) ↔ v = w)
28		f1fun 5037	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐹:A–1-1→B → Fun 𝐹)
29		funopfv 5156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Fun 𝐹 → (⟨𝑎, v⟩ ∈ 𝐹 → (𝐹‘𝑎) = v))
30		funopfv 5156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Fun 𝐹 → (⟨𝑏, w⟩ ∈ 𝐹 → (𝐹‘𝑏) = w))
31	29, 30	anim12d 318	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Fun 𝐹 → ((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) → ((𝐹‘𝑎) = v ∧ (𝐹‘𝑏) = w)))
32	28, 31	syl 14	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹:A–1-1→B → ((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) → ((𝐹‘𝑎) = v ∧ (𝐹‘𝑏) = w)))
33		eqcom 2039	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐹‘𝑎) = v ↔ v = (𝐹‘𝑎))
34	33	biimpi 113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐹‘𝑎) = v → v = (𝐹‘𝑎))
35		eqcom 2039	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐹‘𝑏) = w ↔ w = (𝐹‘𝑏))
36	35	biimpi 113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐹‘𝑏) = w → w = (𝐹‘𝑏))
37	34, 36	eqeqan12d 2052	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝐹‘𝑎) = v ∧ (𝐹‘𝑏) = w) → (v = w ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
38		simpl 102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑎 ∈ A ∧ v ∈ B) → 𝑎 ∈ A)
39		simpl 102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑏 ∈ A ∧ w ∈ B) → 𝑏 ∈ A)
40	38, 39	anim12i 321	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B)) → (𝑎 ∈ A ∧ 𝑏 ∈ A))
41		f1veqaeq 5351	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝐹:A–1-1→B ∧ (𝑎 ∈ A ∧ 𝑏 ∈ A)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
42	40, 41	sylan2 270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐹:A–1-1→B ∧ ((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
43		opeq12 3542	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑎 = 𝑏 ∧ v = w) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)
44	43	ex 108	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑎 = 𝑏 → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))
45	42, 44	syl6 29	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝐹:A–1-1→B ∧ ((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
46	45	com23 72	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐹:A–1-1→B ∧ ((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B))) → (v = w → ((𝐹‘𝑎) = (𝐹‘𝑏) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
47	46	ex 108	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝐹:A–1-1→B → (((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B)) → (v = w → ((𝐹‘𝑎) = (𝐹‘𝑏) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
48	47	com14 82	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐹‘𝑎) = (𝐹‘𝑏) → (((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B)) → (v = w → (𝐹:A–1-1→B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
49	37, 48	syl6bi 152	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐹‘𝑎) = v ∧ (𝐹‘𝑏) = w) → (v = w → (((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B)) → (v = w → (𝐹:A–1-1→B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))))
50	49	com14 82	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (v = w → (v = w → (((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B)) → (((𝐹‘𝑎) = v ∧ (𝐹‘𝑏) = w) → (𝐹:A–1-1→B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))))
51	50	pm2.43i 43	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (v = w → (((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B)) → (((𝐹‘𝑎) = v ∧ (𝐹‘𝑏) = w) → (𝐹:A–1-1→B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
52	51	com14 82	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐹:A–1-1→B → (((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B)) → (((𝐹‘𝑎) = v ∧ (𝐹‘𝑏) = w) → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
53	52	com23 72	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹:A–1-1→B → (((𝐹‘𝑎) = v ∧ (𝐹‘𝑏) = w) → (((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B)) → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
54	32, 53	syld 40	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹:A–1-1→B → ((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) → (((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B)) → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
55	54	com13 74	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B)) → ((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) → (𝐹:A–1-1→B → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
56	55	impcom 116	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) ∧ ((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B))) → (𝐹:A–1-1→B → (v = w → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
57	56	com23 72	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) ∧ ((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B))) → (v = w → (𝐹:A–1-1→B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
58	27, 57	syl5bi 141	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) ∧ ((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B))) → ((2nd ‘⟨𝑎, v⟩) = (2nd ‘⟨𝑏, w⟩) → (𝐹:A–1-1→B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
59	20, 58	sylbid 139	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) ∧ ((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B))) → (((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = ((2nd ↾ 𝐹)‘⟨𝑏, w⟩) → (𝐹:A–1-1→B → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
60	59	com23 72	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) ∧ ((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B))) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = ((2nd ↾ 𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
61	60	ex 108	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹) → (((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B)) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = ((2nd ↾ 𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
62	61	adantl 262	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ⊆ (A × B) ∧ (⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹)) → (((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B)) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = ((2nd ↾ 𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
63	62	com12 27	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎 ∈ A ∧ v ∈ B) ∧ (𝑏 ∈ A ∧ w ∈ B)) → ((𝐹 ⊆ (A × B) ∧ (⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹)) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = ((2nd ↾ 𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
64	63	adantlr 446	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) → ((𝐹 ⊆ (A × B) ∧ (⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹)) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = ((2nd ↾ 𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
65	64	adantr 261	. . . . . . . . . . . . . . 15 ⊢ (((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) ∧ y = ⟨𝑏, w⟩) → ((𝐹 ⊆ (A × B) ∧ (⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹)) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = ((2nd ↾ 𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
66		eleq1 2097	. . . . . . . . . . . . . . . . . 18 ⊢ (x = ⟨𝑎, v⟩ → (x ∈ 𝐹 ↔ ⟨𝑎, v⟩ ∈ 𝐹))
67	66	ad2antlr 458	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) → (x ∈ 𝐹 ↔ ⟨𝑎, v⟩ ∈ 𝐹))
68		eleq1 2097	. . . . . . . . . . . . . . . . 17 ⊢ (y = ⟨𝑏, w⟩ → (y ∈ 𝐹 ↔ ⟨𝑏, w⟩ ∈ 𝐹))
69	67, 68	bi2anan9 538	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) ∧ y = ⟨𝑏, w⟩) → ((x ∈ 𝐹 ∧ y ∈ 𝐹) ↔ (⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹)))
70	69	anbi2d 437	. . . . . . . . . . . . . . 15 ⊢ (((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) ∧ y = ⟨𝑏, w⟩) → ((𝐹 ⊆ (A × B) ∧ (x ∈ 𝐹 ∧ y ∈ 𝐹)) ↔ (𝐹 ⊆ (A × B) ∧ (⟨𝑎, v⟩ ∈ 𝐹 ∧ ⟨𝑏, w⟩ ∈ 𝐹))))
71		fveq2 5121	. . . . . . . . . . . . . . . . . . 19 ⊢ (x = ⟨𝑎, v⟩ → ((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘⟨𝑎, v⟩))
72	71	ad2antlr 458	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) → ((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘⟨𝑎, v⟩))
73		fveq2 5121	. . . . . . . . . . . . . . . . . 18 ⊢ (y = ⟨𝑏, w⟩ → ((2nd ↾ 𝐹)‘y) = ((2nd ↾ 𝐹)‘⟨𝑏, w⟩))
74	72, 73	eqeqan12d 2052	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) ∧ y = ⟨𝑏, w⟩) → (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) ↔ ((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = ((2nd ↾ 𝐹)‘⟨𝑏, w⟩)))
75		simpllr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) ∧ y = ⟨𝑏, w⟩) → x = ⟨𝑎, v⟩)
76		simpr 103	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) ∧ y = ⟨𝑏, w⟩) → y = ⟨𝑏, w⟩)
77	75, 76	eqeq12d 2051	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) ∧ y = ⟨𝑏, w⟩) → (x = y ↔ ⟨𝑎, v⟩ = ⟨𝑏, w⟩))
78	74, 77	imbi12d 223	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) ∧ y = ⟨𝑏, w⟩) → ((((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y) ↔ (((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = ((2nd ↾ 𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩)))
79	78	imbi2d 219	. . . . . . . . . . . . . . 15 ⊢ (((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) ∧ y = ⟨𝑏, w⟩) → ((𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y)) ↔ (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘⟨𝑎, v⟩) = ((2nd ↾ 𝐹)‘⟨𝑏, w⟩) → ⟨𝑎, v⟩ = ⟨𝑏, w⟩))))
80	65, 70, 79	3imtr4d 192	. . . . . . . . . . . . . 14 ⊢ (((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) ∧ y = ⟨𝑏, w⟩) → ((𝐹 ⊆ (A × B) ∧ (x ∈ 𝐹 ∧ y ∈ 𝐹)) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y))))
81	80	ex 108	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) ∧ (𝑏 ∈ A ∧ w ∈ B)) → (y = ⟨𝑏, w⟩ → ((𝐹 ⊆ (A × B) ∧ (x ∈ 𝐹 ∧ y ∈ 𝐹)) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y)))))
82	81	rexlimdvva 2434	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ A ∧ v ∈ B) ∧ x = ⟨𝑎, v⟩) → (∃𝑏 ∈ A ∃w ∈ B y = ⟨𝑏, w⟩ → ((𝐹 ⊆ (A × B) ∧ (x ∈ 𝐹 ∧ y ∈ 𝐹)) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y)))))
83	82	ex 108	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ A ∧ v ∈ B) → (x = ⟨𝑎, v⟩ → (∃𝑏 ∈ A ∃w ∈ B y = ⟨𝑏, w⟩ → ((𝐹 ⊆ (A × B) ∧ (x ∈ 𝐹 ∧ y ∈ 𝐹)) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y))))))
84	83	rexlimivv 2432	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ A ∃v ∈ B x = ⟨𝑎, v⟩ → (∃𝑏 ∈ A ∃w ∈ B y = ⟨𝑏, w⟩ → ((𝐹 ⊆ (A × B) ∧ (x ∈ 𝐹 ∧ y ∈ 𝐹)) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y)))))
85	84	imp 115	. . . . . . . . 9 ⊢ ((∃𝑎 ∈ A ∃v ∈ B x = ⟨𝑎, v⟩ ∧ ∃𝑏 ∈ A ∃w ∈ B y = ⟨𝑏, w⟩) → ((𝐹 ⊆ (A × B) ∧ (x ∈ 𝐹 ∧ y ∈ 𝐹)) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y))))
86	14, 85	mpcom 32	. . . . . . . 8 ⊢ ((𝐹 ⊆ (A × B) ∧ (x ∈ 𝐹 ∧ y ∈ 𝐹)) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y)))
87	86	ex 108	. . . . . . 7 ⊢ (𝐹 ⊆ (A × B) → ((x ∈ 𝐹 ∧ y ∈ 𝐹) → (𝐹:A–1-1→B → (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y))))
88	87	com23 72	. . . . . 6 ⊢ (𝐹 ⊆ (A × B) → (𝐹:A–1-1→B → ((x ∈ 𝐹 ∧ y ∈ 𝐹) → (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y))))
89	7, 88	mpcom 32	. . . . 5 ⊢ (𝐹:A–1-1→B → ((x ∈ 𝐹 ∧ y ∈ 𝐹) → (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y)))
90	89	ralrimivv 2394	. . . 4 ⊢ (𝐹:A–1-1→B → ∀x ∈ 𝐹 ∀y ∈ 𝐹 (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y))
91		dff13 5350	. . . 4 ⊢ ((2nd ↾ 𝐹):𝐹–1-1→B ↔ ((2nd ↾ 𝐹):𝐹⟶B ∧ ∀x ∈ 𝐹 ∀y ∈ 𝐹 (((2nd ↾ 𝐹)‘x) = ((2nd ↾ 𝐹)‘y) → x = y)))
92	5, 90, 91	sylanbrc 394	. . 3 ⊢ (𝐹:A–1-1→B → (2nd ↾ 𝐹):𝐹–1-1→B)
93		df-f1 4850	. . . 4 ⊢ ((2nd ↾ 𝐹):𝐹–1-1→B ↔ ((2nd ↾ 𝐹):𝐹⟶B ∧ Fun ◡(2nd ↾ 𝐹)))
94	93	simprbi 260	. . 3 ⊢ ((2nd ↾ 𝐹):𝐹–1-1→B → Fun ◡(2nd ↾ 𝐹))
95	92, 94	syl 14	. 2 ⊢ (𝐹:A–1-1→B → Fun ◡(2nd ↾ 𝐹))
96		dff1o3 5075	. 2 ⊢ ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 ↔ ((2nd ↾ 𝐹):𝐹–onto→ran 𝐹 ∧ Fun ◡(2nd ↾ 𝐹)))
97	3, 95, 96	sylanbrc 394	1 ⊢ (𝐹:A–1-1→B → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301   ⊆ wss 2911  ⟨cop 3370   × cxp 4286  ^◡ccnv 4287  ran crn 4289   ↾ cres 4290  Fun wfun 4839  ⟶wf 4841  –1-1→wf1 4842  –onto→wfo 4843  –1-1-onto→wf1o 4844  ‘cfv 4845  2^nd c2nd 5708

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-2nd 5710

This theorem is referenced by: (None)