Theorem f1o2ndf1 7285

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 6101	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		fo2ndf 7284	. . 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 7283	. . . . 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 6060	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
7	1, 6	syl 17	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
8		ssel2 3598	. . . . . . . . . . 11 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ (𝐴 × 𝐵))
9		elxp2 5132	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑎 ∈ 𝐴 ∃𝑣 ∈ 𝐵 𝑥 = ⟨𝑎, 𝑣⟩)
10	8, 9	sylib 208	. . . . . . . . . 10 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐹) → ∃𝑎 ∈ 𝐴 ∃𝑣 ∈ 𝐵 𝑥 = ⟨𝑎, 𝑣⟩)
11		ssel2 3598	. . . . . . . . . . 11 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ (𝐴 × 𝐵))
12		elxp2 5132	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐴 × 𝐵) ↔ ∃𝑏 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑦 = ⟨𝑏, 𝑤⟩)
13	11, 12	sylib 208	. . . . . . . . . 10 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ 𝑦 ∈ 𝐹) → ∃𝑏 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑦 = ⟨𝑏, 𝑤⟩)
14	10, 13	anim12dan 882	. . . . . . . . 9 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (∃𝑎 ∈ 𝐴 ∃𝑣 ∈ 𝐵 𝑥 = ⟨𝑎, 𝑣⟩ ∧ ∃𝑏 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑦 = ⟨𝑏, 𝑤⟩))
15		fvres 6207	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨𝑎, 𝑣⟩ ∈ 𝐹 → ((2nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑎, 𝑣⟩))
16	15	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → ((2nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑎, 𝑣⟩))
17	16	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) ∧ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) → ((2nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑎, 𝑣⟩))
18		fvres 6207	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨𝑏, 𝑤⟩ ∈ 𝐹 → ((2nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩) = (2nd ‘⟨𝑏, 𝑤⟩))
19	18	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . 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 2637	. . . . . . . . . . . . . . . . . . . . . 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 3203	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑎 ∈ V
22		vex 3203	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑣 ∈ V
23	21, 22	op2nd 7177	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2nd ‘⟨𝑎, 𝑣⟩) = 𝑣
24		vex 3203	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑏 ∈ V
25		vex 3203	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑤 ∈ V
26	24, 25	op2nd 7177	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2nd ‘⟨𝑏, 𝑤⟩) = 𝑤
27	23, 26	eqeq12i 2636	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((2nd ‘⟨𝑎, 𝑣⟩) = (2nd ‘⟨𝑏, 𝑤⟩) ↔ 𝑣 = 𝑤)
28		f1fun 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)
29		funopfv 6235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Fun 𝐹 → (⟨𝑎, 𝑣⟩ ∈ 𝐹 → (𝐹‘𝑎) = 𝑣))
30		funopfv 6235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Fun 𝐹 → (⟨𝑏, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑏) = 𝑤))
31	29, 30	anim12d 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Fun 𝐹 → ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → ((𝐹‘𝑎) = 𝑣 ∧ (𝐹‘𝑏) = 𝑤)))
32	28, 31	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹:𝐴–1-1→𝐵 → ((⟨𝑎, 𝑣⟩ ∈ 𝐹 ∧ ⟨𝑏, 𝑤⟩ ∈ 𝐹) → ((𝐹‘𝑎) = 𝑣 ∧ (𝐹‘𝑏) = 𝑤)))
33		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐹‘𝑎) = 𝑣 ↔ 𝑣 = (𝐹‘𝑎))
34	33	biimpi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐹‘𝑎) = 𝑣 → 𝑣 = (𝐹‘𝑎))
35		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐹‘𝑏) = 𝑤 ↔ 𝑤 = (𝐹‘𝑏))
36	35	biimpi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐹‘𝑏) = 𝑤 → 𝑤 = (𝐹‘𝑏))
37	34, 36	eqeqan12d 2638	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝐹‘𝑎) = 𝑣 ∧ (𝐹‘𝑏) = 𝑤) → (𝑣 = 𝑤 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
38		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) → 𝑎 ∈ 𝐴)
39		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → 𝑏 ∈ 𝐴)
40	38, 39	anim12i 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴))
41		f1veqaeq 6514	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
42	40, 41	sylan2 491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
43		opeq12 4404	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑎 = 𝑏 ∧ 𝑣 = 𝑤) → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩)
44	43	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑎 = 𝑏 → (𝑣 = 𝑤 → ⟨𝑎, 𝑣⟩ = ⟨𝑏, 𝑤⟩))
45	42, 44	syl6 35	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 96	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 96	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 52	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 96	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 86	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 47	. . . . . . . . . . . . . . . . . . . . . . . . . 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 88	. . . . . . . . . . . . . . . . . . . . . . . . 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 446	. . . . . . . . . . . . . . . . . . . . . . . 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 86	. . . . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . . . 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 230	. . . . . . . . . . . . . . . . . . . . 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 86	. . . . . . . . . . . . . . . . . . . 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 450	. . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . . 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 751	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 2689	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨𝑎, 𝑣⟩ → (𝑥 ∈ 𝐹 ↔ ⟨𝑎, 𝑣⟩ ∈ 𝐹))
67	66	ad2antlr 763	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → (𝑥 ∈ 𝐹 ↔ ⟨𝑎, 𝑣⟩ ∈ 𝐹))
68		eleq1 2689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑏, 𝑤⟩ → (𝑦 ∈ 𝐹 ↔ ⟨𝑏, 𝑤⟩ ∈ 𝐹))
69	67, 68	bi2anan9 917	. . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . 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 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨𝑎, 𝑣⟩ → ((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩))
72	71	ad2antlr 763	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → ((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘⟨𝑎, 𝑣⟩))
73		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑏, 𝑤⟩ → ((2nd ↾ 𝐹)‘𝑦) = ((2nd ↾ 𝐹)‘⟨𝑏, 𝑤⟩))
74	72, 73	eqeqan12d 2638	. . . . . . . . . . . . . . . . 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 799	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → 𝑥 = ⟨𝑎, 𝑣⟩)
76		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑥 = ⟨𝑎, 𝑣⟩) ∧ (𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑦 = ⟨𝑏, 𝑤⟩) → 𝑦 = ⟨𝑏, 𝑤⟩)
77	75, 76	eqeq12d 2637	. . . . . . . . . . . . . . . . 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 334	. . . . . . . . . . . . . . . 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 330	. . . . . . . . . . . . . . 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 283	. . . . . . . . . . . . . 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 450	. . . . . . . . . . . . 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 3038	. . . . . . . . . . . 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 450	. . . . . . . . . . 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 3036	. . . . . . . . . 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 445	. . . . . . . . 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 38	. . . . . . . 8 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝐹:𝐴–1-1→𝐵 → (((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦)))
87	86	ex 450	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝐹:𝐴–1-1→𝐵 → (((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦))))
88	87	com23 86	. . . . . 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 38	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦)))
90	89	ralrimivv 2970	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (((2nd ↾ 𝐹)‘𝑥) = ((2nd ↾ 𝐹)‘𝑦) → 𝑥 = 𝑦))
91		dff13 6512	. . . 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 698	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1→𝐵)
93		df-f1 5893	. . . 4 ⊢ ((2nd ↾ 𝐹):𝐹–1-1→𝐵 ↔ ((2nd ↾ 𝐹):𝐹⟶𝐵 ∧ Fun ◡(2nd ↾ 𝐹)))
94	93	simprbi 480	. . 3 ⊢ ((2nd ↾ 𝐹):𝐹–1-1→𝐵 → Fun ◡(2nd ↾ 𝐹))
95	92, 94	syl 17	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡(2nd ↾ 𝐹))
96		dff1o3 6143	. 2 ⊢ ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 ↔ ((2nd ↾ 𝐹):𝐹–onto→ran 𝐹 ∧ Fun ◡(2nd ↾ 𝐹)))
97	3, 95, 96	sylanbrc 698	1 ⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  ⟨cop 4183   × cxp 5112  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116  Fun wfun 5882  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  2^nd c2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-2nd 7169

This theorem is referenced by:  hashf1rn  13142  hashf1rnOLD  13143