Theorem f1o2ndf1 8127

Description: The 2^nd (second component of an ordered pair) function restricted to a one-to-one function 𝐹 is a one-to-one function from 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 6793	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		fo2ndf 8126	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹–onto→ran 𝐹)
3	1, 2	syl 17	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–onto→ran 𝐹)
4		f2ndf 8125	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)
5	1, 4	syl 17	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)
6		fssxp 6751	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
7	1, 6	syl 17	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
8		ssel2 3971	. . . . . . . . . . 11 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ (𝐴 × 𝐵))
9		elxp2 5702	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑎 ∈ 𝐴 ∃𝑣 ∈ 𝐵 𝑥 = ⟨𝑎, 𝑣⟩)
10	8, 9	sylib 217	. . . . . . . . . 10 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐹) → ∃𝑎 ∈ 𝐴 ∃𝑣 ∈ 𝐵 𝑥 = ⟨𝑎, 𝑣⟩)
11		ssel2 3971	. . . . . . . . . . 11 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ (𝐴 × 𝐵))
12		elxp2 5702	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐴 × 𝐵) ↔ ∃𝑏 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑦 = ⟨𝑏, 𝑤⟩)
13	11, 12	sylib 217	. . . . . . . . . 10 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑦 ∈ 𝐹) → ∃𝑏 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑦 = ⟨𝑏, 𝑤⟩)
14	10, 13	anim12dan 617	. . . . . . . . 9 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (∃𝑎 ∈ 𝐴 ∃𝑣 ∈ 𝐵 𝑥 = ⟨𝑎, 𝑣⟩ ∧ ∃𝑏 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑦 = ⟨𝑏, 𝑤⟩))
15		fvres 6915	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑎, 𝑣⟩ ∈ 𝐹 → ((2nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑎, 𝑣⟩))
16	15	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) → ((2nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑎, 𝑣⟩))
17		fvres 6915	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑏, 𝑤⟩ ∈ 𝐹 → ((2nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩) = (2nd ‘⟨𝑏, 𝑤⟩))
18	17	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) → ((2nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩) = (2nd ‘⟨𝑏, 𝑤⟩))
19	16, 18	eqeq12d 2741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((⟨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⟩)))
20		vex 3465	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑎 ∈ V
21		vex 3465	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑣 ∈ V
22	20, 21	op2nd 8003	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2nd ‘⟨𝑎, 𝑣⟩) = 𝑣
23		vex 3465	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑏 ∈ V
24		vex 3465	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑤 ∈ V
25	23, 24	op2nd 8003	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2nd ‘⟨𝑏, 𝑤⟩) = 𝑤
26	22, 25	eqeq12i 2743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2nd ‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑏, 𝑤⟩) ↔ 𝑣 = 𝑤)
27		f1fun 6795	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)
28		funopfv 6948	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Fun 𝐹 → (⟨𝑎, 𝑣⟩ ∈ 𝐹 → (𝐹‘𝑎) = 𝑣))
29		funopfv 6948	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Fun 𝐹 → (⟨𝑏, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑏) = 𝑤))
30	28, 29	anim12d 607	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Fun 𝐹 → ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → ((𝐹‘𝑎) = 𝑣 ∧ (𝐹‘𝑏) = 𝑤)))
31	27, 30	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹:𝐴–1-1→𝐵 → ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → ((𝐹‘𝑎) = 𝑣 ∧ (𝐹‘𝑏) = 𝑤)))
32		eqcom 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐹‘𝑎) = 𝑣 ↔ 𝑣 = (𝐹‘𝑎))
33	32	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐹‘𝑎) = 𝑣 → 𝑣 = (𝐹‘𝑎))
34		eqcom 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐹‘𝑏) = 𝑤 ↔ 𝑤 = (𝐹‘𝑏))
35	34	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐹‘𝑏) = 𝑤 → 𝑤 = (𝐹‘𝑏))
36	33, 35	eqeqan12d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐹‘𝑎) = 𝑣 ∧ (𝐹‘𝑏) = 𝑤) → (𝑣 = 𝑤 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
37		simpl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) → 𝑎 ∈ 𝐴)
38		simpl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → 𝑏 ∈ 𝐴)
39	37, 38	anim12i 611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴))
40		f1veqaeq 7267	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
41	39, 40	sylan2 591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
42		opeq12 4877	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑎 = 𝑏 ∧ 𝑣 = 𝑤) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)
43	42	ex 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑎 = 𝑏 → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))
44	41, 43	syl6 35	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
45	44	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) → (𝑣 = 𝑤 → ((𝐹‘𝑎) = (𝐹‘𝑏) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
46	45	ex 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (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	com14 96	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐹‘𝑎) = (𝐹‘𝑏) → (((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → (𝑣 = 𝑤 → (𝐹:𝐴–1-1→𝐵 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
48	36, 47	biimtrdi 252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((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⟩)))))
49	48	com14 96	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (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⟩)))))
50	49	pm2.43i 52	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (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	com14 96	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (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⟩))))
52	51	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (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⟩))))
53	31, 52	syld 47	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (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⟩))))
54	53	com13 88	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((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⟩))))
55	54	impcom 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⟨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⟩)))
56	55	com23 86	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⟨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⟩)))
57	26, 56	biimtrid 241	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((⟨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⟩)))
58	19, 57	sylbid 239	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((⟨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⟩)))
59	58	com23 86	. . . . . . . . . . . . . . . . . . 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⟩)))
60	59	ex 411	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨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	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ ((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⟩))))
62	61	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (((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⟩))))
63	62	ad4ant13 749	. . . . . . . . . . . . . . 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⟩))))
64		eleq1 2813	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨𝑎, 𝑣⟩ → (𝑥 ∈ 𝐹 ↔ ⟨𝑎, 𝑣⟩ ∈ 𝐹))
65	64	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → (𝑥 ∈ 𝐹 ↔ ⟨𝑎, 𝑣⟩ ∈ 𝐹))
66		eleq1 2813	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑏, 𝑤⟩ → (𝑦 ∈ 𝐹 ↔ ⟨𝑏, 𝑤⟩ ∈ 𝐹))
67	65, 66	bi2anan9 636	. . . . . . . . . . . . . . . 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)))
68	67	anbi2d 628	. . . . . . . . . . . . . . 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))))
69		fveq2 6896	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨𝑎, 𝑣⟩ → ((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩))
70	69	ad2antlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → ((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩))
71		fveq2 6896	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑏, 𝑤⟩ → ((2nd ↾ 𝐹)‘𝑦) = ((2nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩))
72	70, 71	eqeqan12d 2739	. . . . . . . . . . . . . . . . 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⟩)))
73		simpllr 774	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → 𝑥 = ⟨𝑎, 𝑣⟩)
74		simpr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → 𝑦 = ⟨𝑏, 𝑤⟩)
75	73, 74	eqeq12d 2741	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → (𝑥 = 𝑦 ↔ ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))
76	72, 75	imbi12d 343	. . . . . . . . . . . . . . . 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⟩)))
77	76	imbi2d 339	. . . . . . . . . . . . . . 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⟩))))
78	63, 68, 77	3imtr4d 293	. . . . . . . . . . . . . 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))))
79	78	ex 411	. . . . . . . . . . . . 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)))))
80	79	rexlimdvva 3201	. . . . . . . . . . . 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)))))
81	80	ex 411	. . . . . . . . . . 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))))))
82	81	rexlimivv 3189	. . . . . . . . . 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)))))
83	82	imp 405	. . . . . . . . 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))))
84	14, 83	mpcom 38	. . . . . . . 8 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝐹:𝐴–1-1→𝐵 → (((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦)))
85	84	ex 411	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝐹:𝐴–1-1→𝐵 → (((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦))))
86	85	com23 86	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝐹:𝐴–1-1→𝐵 → ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦))))
87	7, 86	mpcom 38	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦)))
88	87	ralrimivv 3188	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦))
89		dff13 7265	. . . 4 ⊢ ((2nd ↾ 𝐹):𝐹–1-1→𝐵 ↔ ((2nd ↾ 𝐹):𝐹⟶𝐵 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦)))
90	5, 88, 89	sylanbrc 581	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1→𝐵)
91		df-f1 6554	. . . 4 ⊢ ((2nd ↾ 𝐹):𝐹–1-1→𝐵 ↔ ((2nd ↾ 𝐹):𝐹⟶𝐵 ∧ Fun ◡(2nd ↾ 𝐹)))
92	91	simprbi 495	. . 3 ⊢ ((2nd ↾ 𝐹):𝐹–1-1→𝐵 → Fun ◡(2nd ↾ 𝐹))
93	90, 92	syl 17	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡(2nd ↾ 𝐹))
94		dff1o3 6844	. 2 ⊢ ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 ↔ ((2nd ↾ 𝐹):𝐹–onto→ran 𝐹 ∧ Fun ◡(2nd ↾ 𝐹)))
95	3, 93, 94	sylanbrc 581	1 ⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3050  ∃wrex 3059   ⊆ wss 3944  ⟨cop 4636   × cxp 5676  ^◡ccnv 5677  ran crn 5679   ↾ cres 5680  Fun wfun 6543  ⟶wf 6545  –1-1→wf1 6546  –onto→wfo 6547  –1-1-onto→wf1o 6548  ‘cfv 6549  2^nd c2nd 7993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-2nd 7995

This theorem is referenced by:  hashf1rn  14347