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Theorem fo2ndf 8107
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:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹):𝐹–ontoβ†’ran 𝐹)
Proof of Theorem fo2ndf
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffrn 6732 . . 3 (𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴⟢ran 𝐹)
2 f2ndf 8106 . . 3 (𝐹:𝐴⟢ran 𝐹 β†’ (2nd β†Ύ 𝐹):𝐹⟢ran 𝐹)
31, 2syl 17 . 2 (𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹):𝐹⟢ran 𝐹)
4 ffn 6718 . . . . 5 (𝐹:𝐴⟢𝐡 β†’ 𝐹 Fn 𝐴)
5 dffn3 6731 . . . . . 6 (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟢ran 𝐹)
65, 2sylbi 216 . . . . 5 (𝐹 Fn 𝐴 β†’ (2nd β†Ύ 𝐹):𝐹⟢ran 𝐹)
74, 6syl 17 . . . 4 (𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹):𝐹⟢ran 𝐹)
87frnd 6726 . . 3 (𝐹:𝐴⟢𝐡 β†’ ran (2nd β†Ύ 𝐹) βŠ† ran 𝐹)
9 elrn2g 5891 . . . . . 6 (𝑦 ∈ ran 𝐹 β†’ (𝑦 ∈ ran 𝐹 ↔ βˆƒπ‘₯⟨π‘₯, π‘¦βŸ© ∈ 𝐹))
109ibi 267 . . . . 5 (𝑦 ∈ ran 𝐹 β†’ βˆƒπ‘₯⟨π‘₯, π‘¦βŸ© ∈ 𝐹)
11 fvres 6911 . . . . . . . . . 10 (⟨π‘₯, π‘¦βŸ© ∈ 𝐹 β†’ ((2nd β†Ύ 𝐹)β€˜βŸ¨π‘₯, π‘¦βŸ©) = (2nd β€˜βŸ¨π‘₯, π‘¦βŸ©))
1211adantl 483 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ ⟨π‘₯, π‘¦βŸ© ∈ 𝐹) β†’ ((2nd β†Ύ 𝐹)β€˜βŸ¨π‘₯, π‘¦βŸ©) = (2nd β€˜βŸ¨π‘₯, π‘¦βŸ©))
13 vex 3479 . . . . . . . . . 10 π‘₯ ∈ V
14 vex 3479 . . . . . . . . . 10 𝑦 ∈ V
1513, 14op2nd 7984 . . . . . . . . 9 (2nd β€˜βŸ¨π‘₯, π‘¦βŸ©) = 𝑦
1612, 15eqtr2di 2790 . . . . . . . 8 ((𝐹:𝐴⟢𝐡 ∧ ⟨π‘₯, π‘¦βŸ© ∈ 𝐹) β†’ 𝑦 = ((2nd β†Ύ 𝐹)β€˜βŸ¨π‘₯, π‘¦βŸ©))
17 f2ndf 8106 . . . . . . . . . 10 (𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹):𝐹⟢𝐡)
1817ffnd 6719 . . . . . . . . 9 (𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹) Fn 𝐹)
19 fnfvelrn 7083 . . . . . . . . 9 (((2nd β†Ύ 𝐹) Fn 𝐹 ∧ ⟨π‘₯, π‘¦βŸ© ∈ 𝐹) β†’ ((2nd β†Ύ 𝐹)β€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ ran (2nd β†Ύ 𝐹))
2018, 19sylan 581 . . . . . . . 8 ((𝐹:𝐴⟢𝐡 ∧ ⟨π‘₯, π‘¦βŸ© ∈ 𝐹) β†’ ((2nd β†Ύ 𝐹)β€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ ran (2nd β†Ύ 𝐹))
2116, 20eqeltrd 2834 . . . . . . 7 ((𝐹:𝐴⟢𝐡 ∧ ⟨π‘₯, π‘¦βŸ© ∈ 𝐹) β†’ 𝑦 ∈ ran (2nd β†Ύ 𝐹))
2221ex 414 . . . . . 6 (𝐹:𝐴⟢𝐡 β†’ (⟨π‘₯, π‘¦βŸ© ∈ 𝐹 β†’ 𝑦 ∈ ran (2nd β†Ύ 𝐹)))
2322exlimdv 1937 . . . . 5 (𝐹:𝐴⟢𝐡 β†’ (βˆƒπ‘₯⟨π‘₯, π‘¦βŸ© ∈ 𝐹 β†’ 𝑦 ∈ ran (2nd β†Ύ 𝐹)))
2410, 23syl5 34 . . . 4 (𝐹:𝐴⟢𝐡 β†’ (𝑦 ∈ ran 𝐹 β†’ 𝑦 ∈ ran (2nd β†Ύ 𝐹)))
2524ssrdv 3989 . . 3 (𝐹:𝐴⟢𝐡 β†’ ran 𝐹 βŠ† ran (2nd β†Ύ 𝐹))
268, 25eqssd 4000 . 2 (𝐹:𝐴⟢𝐡 β†’ ran (2nd β†Ύ 𝐹) = ran 𝐹)
27 dffo2 6810 . 2 ((2nd β†Ύ 𝐹):𝐹–ontoβ†’ran 𝐹 ↔ ((2nd β†Ύ 𝐹):𝐹⟢ran 𝐹 ∧ ran (2nd β†Ύ 𝐹) = ran 𝐹))
283, 26, 27sylanbrc 584 1 (𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹):𝐹–ontoβ†’ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βŸ¨cop 4635  ran crn 5678   β†Ύ cres 5679   Fn wfn 6539  βŸΆwf 6540  β€“ontoβ†’wfo 6542  β€˜cfv 6544  2nd c2nd 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-2nd 7976
This theorem is referenced by:  f1o2ndf1  8108
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