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Theorem funline 26068
 Description: Show that the Line relationship is a function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funline Line

Proof of Theorem funline
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reeanv 2867 . . . . . 6
2 eqtr3 2454 . . . . . . . . 9
32ad2ant2l 727 . . . . . . . 8
43a1i 11 . . . . . . 7
54rexlimivv 2827 . . . . . 6
61, 5sylbir 205 . . . . 5
76gen2 1556 . . . 4
8 eqeq1 2441 . . . . . . . 8
98anbi2d 685 . . . . . . 7
109rexbidv 2718 . . . . . 6
11 fveq2 5720 . . . . . . . . . 10
1211eleq2d 2502 . . . . . . . . 9
1311eleq2d 2502 . . . . . . . . 9
1412, 133anbi12d 1255 . . . . . . . 8
1514anbi1d 686 . . . . . . 7
1615cbvrexv 2925 . . . . . 6
1710, 16syl6bb 253 . . . . 5
1817mo4 2313 . . . 4
197, 18mpbir 201 . . 3
2019funoprab 6162 . 2
21 df-line2 26063 . . 3 Line
2221funeqi 5466 . 2 Line
2320, 22mpbir 201 1 Line
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936  wal 1549   wceq 1652   wcel 1725  wmo 2281   wne 2598  wrex 2698  cop 3809  ccnv 4869   wfun 5440  cfv 5446  coprab 6074  cec 6895  cn 9992  cee 25819   ccolin 25963  Linecline2 26060 This theorem is referenced by:  fvline  26070 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-iota 5410  df-fun 5448  df-fv 5454  df-oprab 6077  df-line2 26063
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