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Theorem fo2ndf 7822
 Description: The 2nd (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:𝐴𝐵 → (2nd𝐹):𝐹onto→ran 𝐹)

Proof of Theorem fo2ndf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffrn 6511 . . 3 (𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹)
2 f2ndf 7821 . . 3 (𝐹:𝐴⟶ran 𝐹 → (2nd𝐹):𝐹⟶ran 𝐹)
31, 2syl 17 . 2 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹⟶ran 𝐹)
4 ffn 6498 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5 dffn3 6510 . . . . . 6 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
65, 2sylbi 220 . . . . 5 (𝐹 Fn 𝐴 → (2nd𝐹):𝐹⟶ran 𝐹)
74, 6syl 17 . . . 4 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹⟶ran 𝐹)
87frnd 6505 . . 3 (𝐹:𝐴𝐵 → ran (2nd𝐹) ⊆ ran 𝐹)
9 elrn2g 5730 . . . . . 6 (𝑦 ∈ ran 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐹))
109ibi 270 . . . . 5 (𝑦 ∈ ran 𝐹 → ∃𝑥𝑥, 𝑦⟩ ∈ 𝐹)
11 fvres 6677 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ((2nd𝐹)‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
1211adantl 485 . . . . . . . . 9 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd𝐹)‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
13 vex 3413 . . . . . . . . . 10 𝑥 ∈ V
14 vex 3413 . . . . . . . . . 10 𝑦 ∈ V
1513, 14op2nd 7702 . . . . . . . . 9 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
1612, 15eqtr2di 2810 . . . . . . . 8 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 = ((2nd𝐹)‘⟨𝑥, 𝑦⟩))
17 f2ndf 7821 . . . . . . . . . 10 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)
1817ffnd 6499 . . . . . . . . 9 (𝐹:𝐴𝐵 → (2nd𝐹) Fn 𝐹)
19 fnfvelrn 6839 . . . . . . . . 9 (((2nd𝐹) Fn 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd𝐹)‘⟨𝑥, 𝑦⟩) ∈ ran (2nd𝐹))
2018, 19sylan 583 . . . . . . . 8 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd𝐹)‘⟨𝑥, 𝑦⟩) ∈ ran (2nd𝐹))
2116, 20eqeltrd 2852 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ ran (2nd𝐹))
2221ex 416 . . . . . 6 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦 ∈ ran (2nd𝐹)))
2322exlimdv 1934 . . . . 5 (𝐹:𝐴𝐵 → (∃𝑥𝑥, 𝑦⟩ ∈ 𝐹𝑦 ∈ ran (2nd𝐹)))
2410, 23syl5 34 . . . 4 (𝐹:𝐴𝐵 → (𝑦 ∈ ran 𝐹𝑦 ∈ ran (2nd𝐹)))
2524ssrdv 3898 . . 3 (𝐹:𝐴𝐵 → ran 𝐹 ⊆ ran (2nd𝐹))
268, 25eqssd 3909 . 2 (𝐹:𝐴𝐵 → ran (2nd𝐹) = ran 𝐹)
27 dffo2 6580 . 2 ((2nd𝐹):𝐹onto→ran 𝐹 ↔ ((2nd𝐹):𝐹⟶ran 𝐹 ∧ ran (2nd𝐹) = ran 𝐹))
283, 26, 27sylanbrc 586 1 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹onto→ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ⟨cop 4528  ran crn 5525   ↾ cres 5526   Fn wfn 6330  ⟶wf 6331  –onto→wfo 6333  ‘cfv 6335  2nd c2nd 7692 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fo 6341  df-fv 6343  df-2nd 7694 This theorem is referenced by:  f1o2ndf1  7823
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