Theorem funline 32249

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 3107	. . . . . 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 2643	. . . . . . . . 9 ⊢ ((𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ∧ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear ) → 𝑙 = 𝑘)
3	2	ad2ant2l 782	. . . . . . . 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 3036	. . . . . 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 225	. . . . 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 1723	. . . 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 2626	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ↔ 𝑘 = [⟨𝑎, 𝑏⟩]◡ Colinear ))
9	8	anbi2d 740	. . . . . . 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 3052	. . . . . 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 6191	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
12	11	eleq2d 2687	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝑎 ∈ (𝔼‘𝑚)))
13	11	eleq2d 2687	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝑏 ∈ (𝔼‘𝑚)))
14	12, 13	3anbi12d 1400	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ↔ (𝑎 ∈ (𝔼‘𝑚) ∧ 𝑏 ∈ (𝔼‘𝑚) ∧ 𝑎 ≠ 𝑏)))
15	14	anbi1d 741	. . . . . . 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 3172	. . . . . 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 276	. . . . 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 2517	. . . 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 221	. . 3 ⊢ ∃*𝑙∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )
20	19	funoprab 6760	. 2 ⊢ Fun {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
21		df-line2 32244	. . 3 ⊢ Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
22	21	funeqi 5909	. 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 221	1 ⊢ Fun Line

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∃*wmo 2471   ≠ wne 2794  ∃wrex 2913  ⟨cop 4183  ^◡ccnv 5113  Fun wfun 5882  ‘cfv 5888  {coprab 6651  [cec 7740  ℕcn 11020  𝔼cee 25768   Colinear ccolin 32144  Linecline2 32241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-iota 5851  df-fun 5890  df-fv 5896  df-oprab 6654  df-line2 32244

This theorem is referenced by:  fvline  32251