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Theorem fo2ndf 6227
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 5365 . . . 4 (𝐹:𝐴⟢𝐡 β†’ 𝐹 Fn 𝐴)
2 dffn3 5376 . . . 4 (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟢ran 𝐹)
31, 2sylib 122 . . 3 (𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴⟢ran 𝐹)
4 f2ndf 6226 . . 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 5374 . . . 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 4817 . . . . . 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 5539 . . . . . . . . . 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 2740 . . . . . . . . . 10 π‘₯ ∈ V
15 vex 2740 . . . . . . . . . 10 𝑦 ∈ V
1614, 15op2nd 6147 . . . . . . . . 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 6226 . . . . . . . . . 10 (𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹):𝐹⟢𝐡)
19 ffn 5365 . . . . . . . . . 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 5648 . . . . . . . . 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 3161 . . 3 (𝐹:𝐴⟢𝐡 β†’ ran 𝐹 βŠ† ran (2nd β†Ύ 𝐹))
289, 27eqssd 3172 . 2 (𝐹:𝐴⟢𝐡 β†’ ran (2nd β†Ύ 𝐹) = ran 𝐹)
29 dffo2 5442 . 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 3129  βŸ¨cop 3595  ran crn 4627   β†Ύ cres 4628   Fn wfn 5211  βŸΆwf 5212  β€“ontoβ†’wfo 5214  β€˜cfv 5216  2nd c2nd 6139
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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
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 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fo 5222  df-fv 5224  df-2nd 6141
This theorem is referenced by:  f1o2ndf1  6228
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