Theorem funline 35977

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.

Step	Hyp	Ref	Expression
1		reeanv 3217	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ∧ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear )) ↔ (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ∧ ∃𝑚 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear )))
2		eqtr3 2752	. . . . . . . . 9 ⊢ ((𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear ) → 𝑙 = 𝑘)
3	2	ad2ant2l 744	. . . . . . . 8 ⊢ ((((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ∧ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear )) → 𝑙 = 𝑘)
4	3	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ∧ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear )) → 𝑙 = 𝑘))
5	4	rexlimivv 3190	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ∧ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear )) → 𝑙 = 𝑘)
6	1, 5	sylbir 234	. . . . 5 ⊢ ((∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ∧ ∃𝑚 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear )) → 𝑙 = 𝑘)
7	6	gen2 1791	. . . 4 ⊢ ∀𝑙∀𝑘((∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ∧ ∃𝑚 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear )) → 𝑙 = 𝑘)
8		eqeq1 2730	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ↔ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear ))
9	8	anbi2d 628	. . . . . . 7 ⊢ (𝑙 = 𝑘 → (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ↔ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear )))
10	9	rexbidv 3169	. . . . . 6 ⊢ (𝑙 = 𝑘 → (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear )))
11		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
12	11	eleq2d 2812	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝑎 ∈ (𝔼‘𝑚)))
13	11	eleq2d 2812	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝑏 ∈ (𝔼‘𝑚)))
14	12, 13	3anbi12d 1434	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ↔ (𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏)))
15	14	anbi1d 629	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ↔ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear )))
16	15	cbvrexvw 3226	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ↔ ∃𝑚 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear ))
17	10, 16	bitrdi 286	. . . . 5 ⊢ (𝑙 = 𝑘 → (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ↔ ∃𝑚 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear )))
18	17	mo4 2555	. . . 4 ⊢ (∃*𝑙∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ↔ ∀𝑙∀𝑘((∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ∧ ∃𝑚 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏) ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear )) → 𝑙 = 𝑘))
19	7, 18	mpbir 230	. . 3 ⊢ ∃*𝑙∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )
20	19	funoprab 7537	. 2 ⊢ Fun {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
21		df-line2 35972	. . 3 ⊢ Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
22	21	funeqi 6570	. 2 ⊢ (Fun Line ↔ Fun {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )})
23	20, 22	mpbir 230	1 ⊢ Fun Line

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084  ∀wal 1532   = wceq 1534   ∈ wcel 2099  ∃*wmo 2527   ≠ wne 2930  ∃wrex 3060  ⟨cop 4630  ^◡ccnv 5672  Fun wfun 6538  ‘cfv 6544  {coprab 7415  [cec 8722  ℕcn 12256  𝔼cee 28817   Colinear ccolin 35872  Linecline2 35969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-iota 6496  df-fun 6546  df-fv 6552  df-oprab 7418  df-line2 35972

This theorem is referenced by:  fvline  35979