Theorem f1o2ndf1 5849

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 5092	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		fo2ndf 5848	. . 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 5847	. . . . 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 5058	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
7	1, 6	syl 14	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
8		ssel2 2940	. . . . . . . . . . 11 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ (𝐴 × 𝐵))
9		elxp2 4363	. . . . . . . . . . 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 2940	. . . . . . . . . . 11 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ (𝐴 × 𝐵))
12		elxp2 4363	. . . . . . . . . . 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 5198	. . . . . . . . . . . . . . . . . . . . . . . . 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 5198	. . . . . . . . . . . . . . . . . . . . . . . 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 2054	. . . . . . . . . . . . . . . . . . . . . 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 2560	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑎 ∈ V
22		vex 2560	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑣 ∈ V
23	21, 22	op2nd 5774	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2nd ‘⟨𝑎, 𝑣⟩) = 𝑣
24		vex 2560	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑏 ∈ V
25		vex 2560	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑤 ∈ V
26	24, 25	op2nd 5774	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2nd ‘⟨𝑏, 𝑤⟩) = 𝑤
27	23, 26	eqeq12i 2053	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((2nd ‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑏, 𝑤⟩) ↔ 𝑣 = 𝑤)
28		f1fun 5094	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)
29		funopfv 5213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Fun 𝐹 → (⟨𝑎, 𝑣⟩ ∈ 𝐹 → (𝐹‘𝑎) = 𝑣))
30		funopfv 5213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐹‘𝑎) = 𝑣 ↔ 𝑣 = (𝐹‘𝑎))
34	33	biimpi 113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐹‘𝑎) = 𝑣 → 𝑣 = (𝐹‘𝑎))
35		eqcom 2042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐹‘𝑏) = 𝑤 ↔ 𝑤 = (𝐹‘𝑏))
36	35	biimpi 113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐹‘𝑏) = 𝑤 → 𝑤 = (𝐹‘𝑏))
37	34, 36	eqeqan12d 2055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3551	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2100	. . . . . . . . . . . . . . . . . 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 2100	. . . . . . . . . . . . . . . . 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 5178	. . . . . . . . . . . . . . . . . . 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 5178	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑏, 𝑤⟩ → ((2nd ↾ 𝐹)‘𝑦) = ((2nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩))
74	72, 73	eqeqan12d 2055	. . . . . . . . . . . . . . . . 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 2054	. . . . . . . . . . . . . . . . 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 2440	. . . . . . . . . . . 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 2438	. . . . . . . . . 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 2400	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦))
91		dff13 5407	. . . 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 4907	. . . 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 5132	. 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 1243   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307   ⊆ wss 2917  ⟨cop 3378   × cxp 4343  ^◡ccnv 4344  ran crn 4346   ↾ cres 4347  Fun wfun 4896  ⟶wf 4898  –1-1→wf1 4899  –onto→wfo 4900  –1-1-onto→wf1o 4901  ‘cfv 4902  2^nd c2nd 5766

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-2nd 5768

This theorem is referenced by: (None)