Theorem fo2ndf 6363

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		ffn 5469	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		dffn3 5480	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
3	1, 2	sylib 122	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)
4		f2ndf 6362	. . 3 ⊢ (𝐹:𝐴⟶ran 𝐹 → (2nd ↾ 𝐹):𝐹⟶ran 𝐹)
5	3, 4	syl 14	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶ran 𝐹)
6	2, 4	sylbi 121	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (2nd ↾ 𝐹):𝐹⟶ran 𝐹)
7	1, 6	syl 14	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶ran 𝐹)
8		frn 5478	. . . 4 ⊢ ((2nd ↾ 𝐹):𝐹⟶ran 𝐹 → ran (2nd ↾ 𝐹) ⊆ ran 𝐹)
9	7, 8	syl 14	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran (2nd ↾ 𝐹) ⊆ ran 𝐹)
10		elrn2g 4909	. . . . . 6 ⊢ (𝑦 ∈ ran 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹))
11	10	ibi 176	. . . . 5 ⊢ (𝑦 ∈ ran 𝐹 → ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹)
12		fvres 5647	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
13	12	adantl 277	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
14		vex 2802	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
15		vex 2802	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
16	14, 15	op2nd 6283	. . . . . . . . 9 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
17	13, 16	eqtr2di 2279	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 = ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩))
18		f2ndf 6362	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)
19		ffn 5469	. . . . . . . . . 10 ⊢ ((2nd ↾ 𝐹):𝐹⟶𝐵 → (2nd ↾ 𝐹) Fn 𝐹)
20	18, 19	syl 14	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹) Fn 𝐹)
21		fnfvelrn 5760	. . . . . . . . 9 ⊢ (((2nd ↾ 𝐹) Fn 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ 𝐹))
22	20, 21	sylan 283	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd ↾ 𝐹)‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ 𝐹))
23	17, 22	eqeltrd 2306	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ ran (2nd ↾ 𝐹))
24	23	ex 115	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ ran (2nd ↾ 𝐹)))
25	24	exlimdv 1865	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ ran (2nd ↾ 𝐹)))
26	11, 25	syl5 32	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ ran (2nd ↾ 𝐹)))
27	26	ssrdv 3230	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ ran (2nd ↾ 𝐹))
28	9, 27	eqssd 3241	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran (2nd ↾ 𝐹) = ran 𝐹)
29		dffo2 5548	. 2 ⊢ ((2nd ↾ 𝐹):𝐹–onto→ran 𝐹 ↔ ((2nd ↾ 𝐹):𝐹⟶ran 𝐹 ∧ ran (2nd ↾ 𝐹) = ran 𝐹))
30	5, 28, 29	sylanbrc 417	1 ⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹–onto→ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200   ⊆ wss 3197  ⟨cop 3669  ran crn 4717   ↾ cres 4718   Fn wfn 5309  ⟶wf 5310  –onto→wfo 5312  ‘cfv 5314  2^nd c2nd 6275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-fo 5320  df-fv 5322  df-2nd 6277

This theorem is referenced by:  f1o2ndf1  6364