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Theorem fo2ndf 7435
Description: The 2nd (second member of an ordered pair) function restricted to a function 𝐹 is a function of 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 6185 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 dffn3 6194 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
31, 2sylib 208 . . 3 (𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹)
4 f2ndf 7434 . . 3 (𝐹:𝐴⟶ran 𝐹 → (2nd𝐹):𝐹⟶ran 𝐹)
53, 4syl 17 . 2 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹⟶ran 𝐹)
62, 4sylbi 207 . . . . 5 (𝐹 Fn 𝐴 → (2nd𝐹):𝐹⟶ran 𝐹)
71, 6syl 17 . . . 4 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹⟶ran 𝐹)
8 frn 6193 . . . 4 ((2nd𝐹):𝐹⟶ran 𝐹 → ran (2nd𝐹) ⊆ ran 𝐹)
97, 8syl 17 . . 3 (𝐹:𝐴𝐵 → ran (2nd𝐹) ⊆ ran 𝐹)
10 elrn2g 5451 . . . . . 6 (𝑦 ∈ ran 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐹))
1110ibi 256 . . . . 5 (𝑦 ∈ ran 𝐹 → ∃𝑥𝑥, 𝑦⟩ ∈ 𝐹)
12 fvres 6348 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ((2nd𝐹)‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
1312adantl 467 . . . . . . . . 9 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd𝐹)‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
14 vex 3354 . . . . . . . . . 10 𝑥 ∈ V
15 vex 3354 . . . . . . . . . 10 𝑦 ∈ V
1614, 15op2nd 7324 . . . . . . . . 9 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
1713, 16syl6req 2822 . . . . . . . 8 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 = ((2nd𝐹)‘⟨𝑥, 𝑦⟩))
18 f2ndf 7434 . . . . . . . . . 10 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)
19 ffn 6185 . . . . . . . . . 10 ((2nd𝐹):𝐹𝐵 → (2nd𝐹) Fn 𝐹)
2018, 19syl 17 . . . . . . . . 9 (𝐹:𝐴𝐵 → (2nd𝐹) Fn 𝐹)
21 fnfvelrn 6499 . . . . . . . . 9 (((2nd𝐹) Fn 𝐹 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd𝐹)‘⟨𝑥, 𝑦⟩) ∈ ran (2nd𝐹))
2220, 21sylan 569 . . . . . . . 8 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → ((2nd𝐹)‘⟨𝑥, 𝑦⟩) ∈ ran (2nd𝐹))
2317, 22eqeltrd 2850 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ ran (2nd𝐹))
2423ex 397 . . . . . 6 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦 ∈ ran (2nd𝐹)))
2524exlimdv 2013 . . . . 5 (𝐹:𝐴𝐵 → (∃𝑥𝑥, 𝑦⟩ ∈ 𝐹𝑦 ∈ ran (2nd𝐹)))
2611, 25syl5 34 . . . 4 (𝐹:𝐴𝐵 → (𝑦 ∈ ran 𝐹𝑦 ∈ ran (2nd𝐹)))
2726ssrdv 3758 . . 3 (𝐹:𝐴𝐵 → ran 𝐹 ⊆ ran (2nd𝐹))
289, 27eqssd 3769 . 2 (𝐹:𝐴𝐵 → ran (2nd𝐹) = ran 𝐹)
29 dffo2 6260 . 2 ((2nd𝐹):𝐹onto→ran 𝐹 ↔ ((2nd𝐹):𝐹⟶ran 𝐹 ∧ ran (2nd𝐹) = ran 𝐹))
305, 28, 29sylanbrc 572 1 (𝐹:𝐴𝐵 → (2nd𝐹):𝐹onto→ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wex 1852  wcel 2145  wss 3723  cop 4322  ran crn 5250  cres 5251   Fn wfn 6026  wf 6027  ontowfo 6029  cfv 6031  2nd c2nd 7314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-2nd 7316
This theorem is referenced by:  f1o2ndf1  7436
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