Theorem fo2ndf 8130

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

Assertion

Ref	Expression
fo2ndf	⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹–onto→ran 𝐹)

Proof of Theorem fo2ndf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffrn 6721	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)
2		f2ndf 8129	. . 3 ⊢ (𝐹:𝐴⟶ran 𝐹 → (2nd ↾ 𝐹):𝐹⟶ran 𝐹)
3	1, 2	syl 18	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶ran 𝐹)
4		ffn 6707	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		dffn3 6720	. . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
6	5, 2	sylbi 220	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (2nd ↾ 𝐹):𝐹⟶ran 𝐹)
7	4, 6	syl 18	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶ran 𝐹)
8	7	frnd 6716	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran (2nd ↾ 𝐹) ⊆ ran 𝐹)
9		elrn2g 5872	. . . . . 6 ⊢ (𝑦 ∈ ran 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹))
10	9	ibi 270	. . . . 5 ⊢ (𝑦 ∈ ran 𝐹 → ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹)
11		fvres 6902	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
12	11	adantl 487	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
13		vex 3455	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
14		vex 3455	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
15	13, 14	op2nd 8008	. . . . . . . . 9 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
16	12, 15	eqtr2di 2813	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 = ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩))
17		f2ndf 8129	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)
18	17	ffnd 6708	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹) Fn 𝐹)
19		fnfvelrn 7078	. . . . . . . . 9 ⊢ (((2nd ↾ 𝐹) Fn 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ 𝐹))
20	18, 19	sylan 592	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ 𝐹))
21	16, 20	eqeltrd 2861	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ ran (2nd ↾ 𝐹))
22	21	ex 418	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ ran (2nd ↾ 𝐹)))
23	22	exlimdv 1966	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ ran (2nd ↾ 𝐹)))
24	10, 23	syl5 35	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ ran (2nd ↾ 𝐹)))
25	24	ssrdv 3937	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ ran (2nd ↾ 𝐹))
26	8, 25	eqssd 3948	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran (2nd ↾ 𝐹) = ran 𝐹)
27		dffo2 6798	. 2 ⊢ ((2nd ↾ 𝐹):𝐹–onto→ran 𝐹 ↔ ((2nd ↾ 𝐹):𝐹⟶ran 𝐹 ∧ ran (2nd ↾ 𝐹) = ran 𝐹))
28	3, 26, 27	sylanbrc 595	1 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹–onto→ran 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ⟨cop 4590  ran crn 5652   ↾ cres 5653   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537  2^nd c2nd 7998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-2nd 8000

This theorem is used by:  f1o2ndf1  8131