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Theorem fo2ndf 8103
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 6728 . . 3 (𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴⟢ran 𝐹)
2 f2ndf 8102 . . 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 6714 . . . . 5 (𝐹:𝐴⟢𝐡 β†’ 𝐹 Fn 𝐴)
5 dffn3 6727 . . . . . 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 6722 . . 3 (𝐹:𝐴⟢𝐡 β†’ ran (2nd β†Ύ 𝐹) βŠ† ran 𝐹)
9 elrn2g 5888 . . . . . 6 (𝑦 ∈ ran 𝐹 β†’ (𝑦 ∈ ran 𝐹 ↔ βˆƒπ‘₯⟨π‘₯, π‘¦βŸ© ∈ 𝐹))
109ibi 266 . . . . 5 (𝑦 ∈ ran 𝐹 β†’ βˆƒπ‘₯⟨π‘₯, π‘¦βŸ© ∈ 𝐹)
11 fvres 6907 . . . . . . . . . 10 (⟨π‘₯, π‘¦βŸ© ∈ 𝐹 β†’ ((2nd β†Ύ 𝐹)β€˜βŸ¨π‘₯, π‘¦βŸ©) = (2nd β€˜βŸ¨π‘₯, π‘¦βŸ©))
1211adantl 482 . . . . . . . . 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 3478 . . . . . . . . . 10 π‘₯ ∈ V
14 vex 3478 . . . . . . . . . 10 𝑦 ∈ V
1513, 14op2nd 7980 . . . . . . . . 9 (2nd β€˜βŸ¨π‘₯, π‘¦βŸ©) = 𝑦
1612, 15eqtr2di 2789 . . . . . . . 8 ((𝐹:𝐴⟢𝐡 ∧ ⟨π‘₯, π‘¦βŸ© ∈ 𝐹) β†’ 𝑦 = ((2nd β†Ύ 𝐹)β€˜βŸ¨π‘₯, π‘¦βŸ©))
17 f2ndf 8102 . . . . . . . . . 10 (𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹):𝐹⟢𝐡)
1817ffnd 6715 . . . . . . . . 9 (𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹) Fn 𝐹)
19 fnfvelrn 7079 . . . . . . . . 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 580 . . . . . . . 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 2833 . . . . . . 7 ((𝐹:𝐴⟢𝐡 ∧ ⟨π‘₯, π‘¦βŸ© ∈ 𝐹) β†’ 𝑦 ∈ ran (2nd β†Ύ 𝐹))
2221ex 413 . . . . . 6 (𝐹:𝐴⟢𝐡 β†’ (⟨π‘₯, π‘¦βŸ© ∈ 𝐹 β†’ 𝑦 ∈ ran (2nd β†Ύ 𝐹)))
2322exlimdv 1936 . . . . 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 3987 . . 3 (𝐹:𝐴⟢𝐡 β†’ ran 𝐹 βŠ† ran (2nd β†Ύ 𝐹))
268, 25eqssd 3998 . 2 (𝐹:𝐴⟢𝐡 β†’ ran (2nd β†Ύ 𝐹) = ran 𝐹)
27 dffo2 6806 . 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 583 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 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βŸ¨cop 4633  ran crn 5676   β†Ύ cres 5677   Fn wfn 6535  βŸΆwf 6536  β€“ontoβ†’wfo 6538  β€˜cfv 6540  2nd c2nd 7970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-2nd 7972
This theorem is referenced by:  f1o2ndf1  8104
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