Theorem funline 36722

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 3234	. . . . . 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 2782	. . . . . . . . 9 ⊢ ((𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear ) → 𝑙 = 𝑘)
3	2	ad2ant2l 759	. . . . . . . 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 3204	. . . . . 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 238	. . . . 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 1829	. . . 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 2764	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ↔ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear ))
9	8	anbi2d 642	. . . . . . 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 3186	. . . . . 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 6878	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
12	11	eleq2d 2846	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝑎 ∈ (𝔼‘𝑚)))
13	11	eleq2d 2846	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝑏 ∈ (𝔼‘𝑚)))
14	12, 13	3anbi12d 1465	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ↔ (𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏)))
15	14	anbi1d 643	. . . . . . 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 3241	. . . . . 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 290	. . . . 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 2591	. . . 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 234	. . 3 ⊢ ∃*𝑙∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )
20	19	funoprab 7535	. 2 ⊢ Fun {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
21		df-line2 36717	. . 3 ⊢ Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
22	21	funeqi 6554	. 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 234	1 ⊢ Fun Line

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃*wmo 2562   ≠ wne 2955  ∃wrex 3086  ⟨cop 4590  ^◡ccnv 5654  Fun wfun 6527  ‘cfv 6533  {coprab 7414  [cec 8694  ℕcn 12257  𝔼cee 29344   Colinear ccolin 36617  Linecline2 36714

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-iota 6489  df-fun 6535  df-fv 6541  df-oprab 7417  df-line2 36717

This theorem is used by:  fvline  36724