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Theorem funline 34780
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 Fun Line

Proof of Theorem funline
Dummy variables 𝑎 𝑏 𝑘 𝑙 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reeanv 3216 . . . . . 6 (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ∧ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear )) ↔ (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ∧ ∃𝑚 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear )))
2 eqtr3 2759 . . . . . . . . 9 ((𝑙 = [⟨𝑎, 𝑏⟩] Colinear ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear ) → 𝑙 = 𝑘)
32ad2ant2l 745 . . . . . . . 8 ((((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ∧ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear )) → 𝑙 = 𝑘)
43a1i 11 . . . . . . 7 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ∧ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear )) → 𝑙 = 𝑘))
54rexlimivv 3193 . . . . . 6 (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ∧ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear )) → 𝑙 = 𝑘)
61, 5sylbir 234 . . . . 5 ((∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ∧ ∃𝑚 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear )) → 𝑙 = 𝑘)
76gen2 1799 . . . 4 𝑙𝑘((∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ∧ ∃𝑚 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear )) → 𝑙 = 𝑘)
8 eqeq1 2737 . . . . . . . 8 (𝑙 = 𝑘 → (𝑙 = [⟨𝑎, 𝑏⟩] Colinear ↔ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear ))
98anbi2d 630 . . . . . . 7 (𝑙 = 𝑘 → (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ↔ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear )))
109rexbidv 3172 . . . . . 6 (𝑙 = 𝑘 → (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear )))
11 fveq2 6846 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
1211eleq2d 2820 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝑎 ∈ (𝔼‘𝑚)))
1311eleq2d 2820 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝑏 ∈ (𝔼‘𝑚)))
1412, 133anbi12d 1438 . . . . . . . 8 (𝑛 = 𝑚 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ↔ (𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎𝑏)))
1514anbi1d 631 . . . . . . 7 (𝑛 = 𝑚 → (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear ) ↔ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear )))
1615cbvrexvw 3225 . . . . . 6 (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear ) ↔ ∃𝑚 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear ))
1710, 16bitrdi 287 . . . . 5 (𝑙 = 𝑘 → (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ↔ ∃𝑚 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear )))
1817mo4 2561 . . . 4 (∃*𝑙𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ↔ ∀𝑙𝑘((∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ∧ ∃𝑚 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩] Colinear )) → 𝑙 = 𝑘))
197, 18mpbir 230 . . 3 ∃*𝑙𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )
2019funoprab 7482 . 2 Fun {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )}
21 df-line2 34775 . . 3 Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )}
2221funeqi 6526 . 2 (Fun Line ↔ Fun {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )})
2320, 22mpbir 230 1 Fun Line
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wal 1540   = wceq 1542  wcel 2107  ∃*wmo 2533  wne 2940  wrex 3070  cop 4596  ccnv 5636  Fun wfun 6494  cfv 6500  {coprab 7362  [cec 8652  cn 12161  𝔼cee 27886   Colinear ccolin 34675  Linecline2 34772
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 5260  ax-nul 5267  ax-pr 5388
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-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-iota 6452  df-fun 6502  df-fv 6508  df-oprab 7365  df-line2 34775
This theorem is referenced by:  fvline  34782
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