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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  ontowfo 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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