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Theorem fo2ndf 6092
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 ffn 5242 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 dffn3 5253 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
31, 2sylib 121 . . 3 (𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹)
4 f2ndf 6091 . . 3 (𝐹:𝐴⟶ran 𝐹 → (2nd𝐹):𝐹⟶ran 𝐹)
53, 4syl 14 . 2 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹⟶ran 𝐹)
62, 4sylbi 120 . . . . 5 (𝐹 Fn 𝐴 → (2nd𝐹):𝐹⟶ran 𝐹)
71, 6syl 14 . . . 4 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹⟶ran 𝐹)
8 frn 5251 . . . 4 ((2nd𝐹):𝐹⟶ran 𝐹 → ran (2nd𝐹) ⊆ ran 𝐹)
97, 8syl 14 . . 3 (𝐹:𝐴𝐵 → ran (2nd𝐹) ⊆ ran 𝐹)
10 elrn2g 4699 . . . . . 6 (𝑦 ∈ ran 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐹))
1110ibi 175 . . . . 5 (𝑦 ∈ ran 𝐹 → ∃𝑥𝑥, 𝑦⟩ ∈ 𝐹)
12 fvres 5413 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ((2nd𝐹)‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
1312adantl 275 . . . . . . . . 9 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd𝐹)‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
14 vex 2663 . . . . . . . . . 10 𝑥 ∈ V
15 vex 2663 . . . . . . . . . 10 𝑦 ∈ V
1614, 15op2nd 6013 . . . . . . . . 9 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
1713, 16syl6req 2167 . . . . . . . 8 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 = ((2nd𝐹)‘⟨𝑥, 𝑦⟩))
18 f2ndf 6091 . . . . . . . . . 10 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)
19 ffn 5242 . . . . . . . . . 10 ((2nd𝐹):𝐹𝐵 → (2nd𝐹) Fn 𝐹)
2018, 19syl 14 . . . . . . . . 9 (𝐹:𝐴𝐵 → (2nd𝐹) Fn 𝐹)
21 fnfvelrn 5520 . . . . . . . . 9 (((2nd𝐹) Fn 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd𝐹)‘⟨𝑥, 𝑦⟩) ∈ ran (2nd𝐹))
2220, 21sylan 281 . . . . . . . 8 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd𝐹)‘⟨𝑥, 𝑦⟩) ∈ ran (2nd𝐹))
2317, 22eqeltrd 2194 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ ran (2nd𝐹))
2423ex 114 . . . . . 6 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦 ∈ ran (2nd𝐹)))
2524exlimdv 1775 . . . . 5 (𝐹:𝐴𝐵 → (∃𝑥𝑥, 𝑦⟩ ∈ 𝐹𝑦 ∈ ran (2nd𝐹)))
2611, 25syl5 32 . . . 4 (𝐹:𝐴𝐵 → (𝑦 ∈ ran 𝐹𝑦 ∈ ran (2nd𝐹)))
2726ssrdv 3073 . . 3 (𝐹:𝐴𝐵 → ran 𝐹 ⊆ ran (2nd𝐹))
289, 27eqssd 3084 . 2 (𝐹:𝐴𝐵 → ran (2nd𝐹) = ran 𝐹)
29 dffo2 5319 . 2 ((2nd𝐹):𝐹onto→ran 𝐹 ↔ ((2nd𝐹):𝐹⟶ran 𝐹 ∧ ran (2nd𝐹) = ran 𝐹))
305, 28, 29sylanbrc 413 1 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹onto→ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wex 1453  wcel 1465  wss 3041  cop 3500  ran crn 4510  cres 4511   Fn wfn 5088  wf 5089  ontowfo 5091  cfv 5093  2nd c2nd 6005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101  df-2nd 6007
This theorem is referenced by:  f1o2ndf1  6093
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