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Theorem fo2ndf 6230
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 ffn 5367 . . . 4 (𝐹:𝐴⟢𝐡 β†’ 𝐹 Fn 𝐴)
2 dffn3 5378 . . . 4 (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟢ran 𝐹)
31, 2sylib 122 . . 3 (𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴⟢ran 𝐹)
4 f2ndf 6229 . . 3 (𝐹:𝐴⟢ran 𝐹 β†’ (2nd β†Ύ 𝐹):𝐹⟢ran 𝐹)
53, 4syl 14 . 2 (𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹):𝐹⟢ran 𝐹)
62, 4sylbi 121 . . . . 5 (𝐹 Fn 𝐴 β†’ (2nd β†Ύ 𝐹):𝐹⟢ran 𝐹)
71, 6syl 14 . . . 4 (𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹):𝐹⟢ran 𝐹)
8 frn 5376 . . . 4 ((2nd β†Ύ 𝐹):𝐹⟢ran 𝐹 β†’ ran (2nd β†Ύ 𝐹) βŠ† ran 𝐹)
97, 8syl 14 . . 3 (𝐹:𝐴⟢𝐡 β†’ ran (2nd β†Ύ 𝐹) βŠ† ran 𝐹)
10 elrn2g 4819 . . . . . 6 (𝑦 ∈ ran 𝐹 β†’ (𝑦 ∈ ran 𝐹 ↔ βˆƒπ‘₯⟨π‘₯, π‘¦βŸ© ∈ 𝐹))
1110ibi 176 . . . . 5 (𝑦 ∈ ran 𝐹 β†’ βˆƒπ‘₯⟨π‘₯, π‘¦βŸ© ∈ 𝐹)
12 fvres 5541 . . . . . . . . . 10 (⟨π‘₯, π‘¦βŸ© ∈ 𝐹 β†’ ((2nd β†Ύ 𝐹)β€˜βŸ¨π‘₯, π‘¦βŸ©) = (2nd β€˜βŸ¨π‘₯, π‘¦βŸ©))
1312adantl 277 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ ⟨π‘₯, π‘¦βŸ© ∈ 𝐹) β†’ ((2nd β†Ύ 𝐹)β€˜βŸ¨π‘₯, π‘¦βŸ©) = (2nd β€˜βŸ¨π‘₯, π‘¦βŸ©))
14 vex 2742 . . . . . . . . . 10 π‘₯ ∈ V
15 vex 2742 . . . . . . . . . 10 𝑦 ∈ V
1614, 15op2nd 6150 . . . . . . . . 9 (2nd β€˜βŸ¨π‘₯, π‘¦βŸ©) = 𝑦
1713, 16eqtr2di 2227 . . . . . . . 8 ((𝐹:𝐴⟢𝐡 ∧ ⟨π‘₯, π‘¦βŸ© ∈ 𝐹) β†’ 𝑦 = ((2nd β†Ύ 𝐹)β€˜βŸ¨π‘₯, π‘¦βŸ©))
18 f2ndf 6229 . . . . . . . . . 10 (𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹):𝐹⟢𝐡)
19 ffn 5367 . . . . . . . . . 10 ((2nd β†Ύ 𝐹):𝐹⟢𝐡 β†’ (2nd β†Ύ 𝐹) Fn 𝐹)
2018, 19syl 14 . . . . . . . . 9 (𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹) Fn 𝐹)
21 fnfvelrn 5650 . . . . . . . . 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))
2220, 21sylan 283 . . . . . . . 8 ((𝐹:𝐴⟢𝐡 ∧ ⟨π‘₯, π‘¦βŸ© ∈ 𝐹) β†’ ((2nd β†Ύ 𝐹)β€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ ran (2nd β†Ύ 𝐹))
2317, 22eqeltrd 2254 . . . . . . 7 ((𝐹:𝐴⟢𝐡 ∧ ⟨π‘₯, π‘¦βŸ© ∈ 𝐹) β†’ 𝑦 ∈ ran (2nd β†Ύ 𝐹))
2423ex 115 . . . . . 6 (𝐹:𝐴⟢𝐡 β†’ (⟨π‘₯, π‘¦βŸ© ∈ 𝐹 β†’ 𝑦 ∈ ran (2nd β†Ύ 𝐹)))
2524exlimdv 1819 . . . . 5 (𝐹:𝐴⟢𝐡 β†’ (βˆƒπ‘₯⟨π‘₯, π‘¦βŸ© ∈ 𝐹 β†’ 𝑦 ∈ ran (2nd β†Ύ 𝐹)))
2611, 25syl5 32 . . . 4 (𝐹:𝐴⟢𝐡 β†’ (𝑦 ∈ ran 𝐹 β†’ 𝑦 ∈ ran (2nd β†Ύ 𝐹)))
2726ssrdv 3163 . . 3 (𝐹:𝐴⟢𝐡 β†’ ran 𝐹 βŠ† ran (2nd β†Ύ 𝐹))
289, 27eqssd 3174 . 2 (𝐹:𝐴⟢𝐡 β†’ ran (2nd β†Ύ 𝐹) = ran 𝐹)
29 dffo2 5444 . 2 ((2nd β†Ύ 𝐹):𝐹–ontoβ†’ran 𝐹 ↔ ((2nd β†Ύ 𝐹):𝐹⟢ran 𝐹 ∧ ran (2nd β†Ύ 𝐹) = ran 𝐹))
305, 28, 29sylanbrc 417 1 (𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹):𝐹–ontoβ†’ran 𝐹)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1353  βˆƒwex 1492   ∈ wcel 2148   βŠ† wss 3131  βŸ¨cop 3597  ran crn 4629   β†Ύ cres 4630   Fn wfn 5213  βŸΆwf 5214  β€“ontoβ†’wfo 5216  β€˜cfv 5218  2nd c2nd 6142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fo 5224  df-fv 5226  df-2nd 6144
This theorem is referenced by:  f1o2ndf1  6231
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