Theorem fo2ndf 8135

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 6724	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)
2		f2ndf 8134	. . 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 6710	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		dffn3 6723	. . . . . 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 6719	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran (2nd ↾ 𝐹) ⊆ ran 𝐹)
9		elrn2g 5871	. . . . . 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 6905	. . . . . . . . . 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 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
14		vex 3454	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
15	13, 14	op2nd 8011	. . . . . . . . 9 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
16	12, 15	eqtr2di 2812	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 = ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩))
17		f2ndf 8134	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)
18	17	ffnd 6711	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹) Fn 𝐹)
19		fnfvelrn 7081	. . . . . . . . 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 2860	. . . . . . 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 1965	. . . . 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 3914	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ ran (2nd ↾ 𝐹))
26	8, 25	eqssd 3925	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran (2nd ↾ 𝐹) = ran 𝐹)
27		dffo2 6801	. 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 1569  ∃wex 1811   ∈ wcel 2144  ⟨cop 4834  ran crn 5651   ↾ cres 5652   Fn wfn 6534  ⟶wf 6535  –onto→wfo 6537  ‘cfv 6539  2^nd c2nd 8001

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fo 6545  df-fv 6547  df-2nd 8003

This theorem is used by:  f1o2ndf1  8136