Theorem funline 32568

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 3295	. . . . . 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 2827	. . . . . . . . 9 ⊢ ((𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear ) → 𝑙 = 𝑘)
3	2	ad2ant2l 743	. . . . . . . 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 3224	. . . . . 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 226	. . . . 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 1878	. . . 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 2810	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ↔ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear ))
9	8	anbi2d 616	. . . . . . 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 3240	. . . . . 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 6404	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
12	11	eleq2d 2871	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝑎 ∈ (𝔼‘𝑚)))
13	11	eleq2d 2871	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝑏 ∈ (𝔼‘𝑚)))
14	12, 13	3anbi12d 1554	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ↔ (𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏)))
15	14	anbi1d 617	. . . . . . 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	cbvrexv 3361	. . . . . 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	syl6bb 278	. . . . 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 2680	. . . 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 222	. . 3 ⊢ ∃*𝑙∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )
20	19	funoprab 6986	. 2 ⊢ Fun {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
21		df-line2 32563	. . 3 ⊢ Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
22	21	funeqi 6118	. 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 222	1 ⊢ Fun Line

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1100  ∀wal 1635   = wceq 1637   ∈ wcel 2156  ∃*wmo 2631   ≠ wne 2978  ∃wrex 3097  ⟨cop 4376  ^◡ccnv 5310  Fun wfun 6091  ‘cfv 6097  {coprab 6871  [cec 7973  ℕcn 11301  𝔼cee 25981   Colinear ccolin 32463  Linecline2 32560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-iota 6060  df-fun 6099  df-fv 6105  df-oprab 6874  df-line2 32563

This theorem is referenced by:  fvline  32570