Theorem fo2ndf 8112

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 6717	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)
2		f2ndf 8111	. . 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 6703	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		dffn3 6716	. . . . . 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 6712	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran (2nd ↾ 𝐹) ⊆ ran 𝐹)
9		elrn2g 5878	. . . . . 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 6898	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
12	11	adantl 486	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
13		vex 3467	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
14		vex 3467	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
15	13, 14	op2nd 7991	. . . . . . . . 9 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
16	12, 15	eqtr2di 2821	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 = ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩))
17		f2ndf 8111	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)
18	17	ffnd 6704	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹) Fn 𝐹)
19		fnfvelrn 7073	. . . . . . . . 9 ⊢ (((2nd ↾ 𝐹) Fn 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ 𝐹))
20	18, 19	sylan 591	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ 𝐹))
21	16, 20	eqeltrd 2869	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ ran (2nd ↾ 𝐹))
22	21	ex 417	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ ran (2nd ↾ 𝐹)))
23	22	exlimdv 1960	. . . . 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 3951	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ ran (2nd ↾ 𝐹))
26	8, 25	eqssd 3962	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran (2nd ↾ 𝐹) = ran 𝐹)
27		dffo2 6794	. 2 ⊢ ((2nd ↾ 𝐹):𝐹–onto→ran 𝐹 ↔ ((2nd ↾ 𝐹):𝐹⟶ran 𝐹 ∧ ran (2nd ↾ 𝐹) = ran 𝐹))
28	3, 26, 27	sylanbrc 594	1 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹–onto→ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ⟨cop 4597  ran crn 5660   ↾ cres 5661   Fn wfn 6528  ⟶wf 6529  –onto→wfo 6531  ‘cfv 6533  2^nd c2nd 7981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-2nd 7983

This theorem is referenced by:  f1o2ndf1  8113