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Theorem fo2ndf 6124
 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 5272 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 dffn3 5283 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
31, 2sylib 121 . . 3 (𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹)
4 f2ndf 6123 . . 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 5281 . . . 4 ((2nd𝐹):𝐹⟶ran 𝐹 → ran (2nd𝐹) ⊆ ran 𝐹)
97, 8syl 14 . . 3 (𝐹:𝐴𝐵 → ran (2nd𝐹) ⊆ ran 𝐹)
10 elrn2g 4729 . . . . . 6 (𝑦 ∈ ran 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐹))
1110ibi 175 . . . . 5 (𝑦 ∈ ran 𝐹 → ∃𝑥𝑥, 𝑦⟩ ∈ 𝐹)
12 fvres 5445 . . . . . . . . . 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 2689 . . . . . . . . . 10 𝑥 ∈ V
15 vex 2689 . . . . . . . . . 10 𝑦 ∈ V
1614, 15op2nd 6045 . . . . . . . . 9 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
1713, 16syl6req 2189 . . . . . . . 8 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 = ((2nd𝐹)‘⟨𝑥, 𝑦⟩))
18 f2ndf 6123 . . . . . . . . . 10 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)
19 ffn 5272 . . . . . . . . . 10 ((2nd𝐹):𝐹𝐵 → (2nd𝐹) Fn 𝐹)
2018, 19syl 14 . . . . . . . . 9 (𝐹:𝐴𝐵 → (2nd𝐹) Fn 𝐹)
21 fnfvelrn 5552 . . . . . . . . 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 2216 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ ran (2nd𝐹))
2423ex 114 . . . . . 6 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦 ∈ ran (2nd𝐹)))
2524exlimdv 1791 . . . . 5 (𝐹:𝐴𝐵 → (∃𝑥𝑥, 𝑦⟩ ∈ 𝐹𝑦 ∈ ran (2nd𝐹)))
2611, 25syl5 32 . . . 4 (𝐹:𝐴𝐵 → (𝑦 ∈ ran 𝐹𝑦 ∈ ran (2nd𝐹)))
2726ssrdv 3103 . . 3 (𝐹:𝐴𝐵 → ran 𝐹 ⊆ ran (2nd𝐹))
289, 27eqssd 3114 . 2 (𝐹:𝐴𝐵 → ran (2nd𝐹) = ran 𝐹)
29 dffo2 5349 . 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 1331  ∃wex 1468   ∈ wcel 1480   ⊆ wss 3071  ⟨cop 3530  ran crn 4540   ↾ cres 4541   Fn wfn 5118  ⟶wf 5119  –onto→wfo 5121  ‘cfv 5123  2nd c2nd 6037 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131  df-2nd 6039 This theorem is referenced by:  f1o2ndf1  6125
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