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Theorem f1o2ndf1 8108

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 → (2^nd ↾ 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 6781	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		fo2ndf 8107	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (2^nd ↾ 𝐹):𝐹–onto→ran 𝐹)
3	1, 2	syl 17	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → (2^nd ↾ 𝐹):𝐹–onto→ran 𝐹)
4		f2ndf 8106	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (2^nd ↾ 𝐹):𝐹⟶𝐵)
5	1, 4	syl 17	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (2^nd ↾ 𝐹):𝐹⟶𝐵)
6		fssxp 6739	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
7	1, 6	syl 17	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
8		ssel2 3972	. . . . . . . . . . 11 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ (𝐴 × 𝐵))
9		elxp2 5693	. . . . . . . . . . 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 3972	. . . . . . . . . . 11 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ (𝐴 × 𝐵))
12		elxp2 5693	. . . . . . . . . . 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 618	. . . . . . . . 9 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (∃𝑎 ∈ 𝐴 ∃𝑣 ∈ 𝐵 𝑥 = ⟨𝑎, 𝑣⟩ ∧ ∃𝑏 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑦 = ⟨𝑏, 𝑤⟩))
15		fvres 6904	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑎, 𝑣⟩ ∈ 𝐹 → ((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = (2^nd ‘⟨𝑎, 𝑣⟩))
16	15	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) → ((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = (2^nd ‘⟨𝑎, 𝑣⟩))
17		fvres 6904	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑏, 𝑤⟩ ∈ 𝐹 → ((2^nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩) = (2^nd ‘⟨𝑏, 𝑤⟩))
18	17	ad2antlr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) → ((2^nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩) = (2^nd ‘⟨𝑏, 𝑤⟩))
19	16, 18	eqeq12d 2742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) → (((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = ((2^nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩) ↔ (2^nd ‘⟨𝑎, 𝑣⟩) = (2^nd ‘⟨𝑏, 𝑤⟩)))
20		vex 3472	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑎 ∈ V
21		vex 3472	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑣 ∈ V
22	20, 21	op2nd 7983	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2^nd ‘⟨𝑎, 𝑣⟩) = 𝑣
23		vex 3472	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑏 ∈ V
24		vex 3472	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑤 ∈ V
25	23, 24	op2nd 7983	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2^nd ‘⟨𝑏, 𝑤⟩) = 𝑤
26	22, 25	eqeq12i 2744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2^nd ‘⟨𝑎, 𝑣⟩) = (2^nd ‘⟨𝑏, 𝑤⟩) ↔ 𝑣 = 𝑤)
27		f1fun 6783	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)
28		funopfv 6937	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Fun 𝐹 → (⟨𝑎, 𝑣⟩ ∈ 𝐹 → (𝐹‘𝑎) = 𝑣))
29		funopfv 6937	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Fun 𝐹 → (⟨𝑏, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑏) = 𝑤))
30	28, 29	anim12d 608	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐹‘𝑎) = 𝑣 ↔ 𝑣 = (𝐹‘𝑎))
33	32	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐹‘𝑎) = 𝑣 → 𝑣 = (𝐹‘𝑎))
34		eqcom 2733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐹‘𝑏) = 𝑤 ↔ 𝑤 = (𝐹‘𝑏))
35	34	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐹‘𝑏) = 𝑤 → 𝑤 = (𝐹‘𝑏))
36	33, 35	eqeqan12d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐹‘𝑎) = 𝑣 ∧ (𝐹‘𝑏) = 𝑤) → (𝑣 = 𝑤 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
37		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) → 𝑎 ∈ 𝐴)
38		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → 𝑏 ∈ 𝐴)
39	37, 38	anim12i 612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴))
40		f1veqaeq 7252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
41	39, 40	sylan2 592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
42		opeq12 4870	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑎 = 𝑏 ∧ 𝑣 = 𝑤) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)
43	42	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . . . . . 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))) → ((2^nd ‘⟨𝑎, 𝑣⟩) = (2^nd ‘⟨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))) → (((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = ((2^nd ↾ 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 → (((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = ((2^nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
60	59	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → (((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → (𝐹:𝐴–1-1→𝐵 → (((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = ((2^nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
61	60	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹)) → (((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → (𝐹:𝐴–1-1→𝐵 → (((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = ((2^nd ↾ 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 → (((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = ((2^nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
63	62	ad4ant13 748	. . . . . . . . . . . . . . 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 → (((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = ((2^nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
64		eleq1 2815	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨𝑎, 𝑣⟩ → (𝑥 ∈ 𝐹 ↔ ⟨𝑎, 𝑣⟩ ∈ 𝐹))
65	64	ad2antlr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → (𝑥 ∈ 𝐹 ↔ ⟨𝑎, 𝑣⟩ ∈ 𝐹))
66		eleq1 2815	. . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨𝑎, 𝑣⟩ → ((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩))
70	69	ad2antlr 724	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → ((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩))
71		fveq2 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑏, 𝑤⟩ → ((2^nd ↾ 𝐹)‘𝑦) = ((2^nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩))
72	70, 71	eqeqan12d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) ↔ ((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = ((2^nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩)))
73		simpllr 773	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → 𝑥 = ⟨𝑎, 𝑣⟩)
74		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → 𝑦 = ⟨𝑏, 𝑤⟩)
75	73, 74	eqeq12d 2742	. . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → ((((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦) ↔ (((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = ((2^nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)))
77	76	imbi2d 340	. . . . . . . . . . . . . . 15 ⊢ (((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → ((𝐹:𝐴–1-1→𝐵 → (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦)) ↔ (𝐹:𝐴–1-1→𝐵 → (((2^nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = ((2^nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))))
78	63, 68, 77	3imtr4d 294	. . . . . . . . . . . . . 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 → (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦))))
79	78	ex 412	. . . . . . . . . . . . 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 → (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦)))))
80	79	rexlimdvva 3205	. . . . . . . . . . . 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 → (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦)))))
81	80	ex 412	. . . . . . . . . . 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 → (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦))))))
82	81	rexlimivv 3193	. . . . . . . . . 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 → (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦)))))
83	82	imp 406	. . . . . . . . 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 → (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦))))
84	14, 83	mpcom 38	. . . . . . . 8 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝐹:𝐴–1-1→𝐵 → (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦)))
85	84	ex 412	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝐹:𝐴–1-1→𝐵 → (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦))))
86	85	com23 86	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝐹:𝐴–1-1→𝐵 → ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦))))
87	7, 86	mpcom 38	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦)))
88	87	ralrimivv 3192	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦))
89		dff13 7250	. . . 4 ⊢ ((2^nd ↾ 𝐹):𝐹–1-1→𝐵 ↔ ((2^nd ↾ 𝐹):𝐹⟶𝐵 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (((2^nd ↾ 𝐹)‘𝑥) = ((2^nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦)))
90	5, 88, 89	sylanbrc 582	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (2^nd ↾ 𝐹):𝐹–1-1→𝐵)
91		df-f1 6542	. . . 4 ⊢ ((2^nd ↾ 𝐹):𝐹–1-1→𝐵 ↔ ((2^nd ↾ 𝐹):𝐹⟶𝐵 ∧ Fun ^◡(2^nd ↾ 𝐹)))
92	91	simprbi 496	. . 3 ⊢ ((2^nd ↾ 𝐹):𝐹–1-1→𝐵 → Fun ^◡(2^nd ↾ 𝐹))
93	90, 92	syl 17	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ^◡(2^nd ↾ 𝐹))
94		dff1o3 6833	. 2 ⊢ ((2^nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 ↔ ((2^nd ↾ 𝐹):𝐹–onto→ran 𝐹 ∧ Fun ^◡(2^nd ↾ 𝐹)))
95	3, 93, 94	sylanbrc 582	1 ⊢ (𝐹:𝐴–1-1→𝐵 → (2^nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 ⊆ wss 3943 ⟨cop 4629 × cxp 5667 ^◡ccnv 5668 ran crn 5670 ↾ cres 5671 Fun wfun 6531 ⟶wf 6533 –1-1→wf1 6534 –onto→wfo 6535 –1-1-onto→wf1o 6536 ‘cfv 6537 2^nd c2nd 7973

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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-2nd 7975

This theorem is referenced by: hashf1rn 14317

W3C validator