MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1dom3fv3dif Structured version   Visualization version   GIF version

Theorem f1dom3fv3dif 6480
Description: The function values for a 1-1 function from a set with three different elements are different. (Contributed by AV, 20-Mar-2019.)
Hypotheses
Ref Expression
f1dom3fv3dif.v (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
f1dom3fv3dif.n (𝜑 → (𝐴𝐵𝐴𝐶𝐵𝐶))
f1dom3fv3dif.f (𝜑𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅)
Assertion
Ref Expression
f1dom3fv3dif (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐵) ∧ (𝐹𝐴) ≠ (𝐹𝐶) ∧ (𝐹𝐵) ≠ (𝐹𝐶)))

Proof of Theorem f1dom3fv3dif
StepHypRef Expression
1 f1dom3fv3dif.n . . . 4 (𝜑 → (𝐴𝐵𝐴𝐶𝐵𝐶))
21simp1d 1071 . . 3 (𝜑𝐴𝐵)
3 f1dom3fv3dif.f . . . . 5 (𝜑𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅)
4 eqidd 2627 . . . . . . 7 (𝜑𝐴 = 𝐴)
543mix1d 1234 . . . . . 6 (𝜑 → (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶))
6 f1dom3fv3dif.v . . . . . . . 8 (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
76simp1d 1071 . . . . . . 7 (𝜑𝐴𝑋)
8 eltpg 4203 . . . . . . 7 (𝐴𝑋 → (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶)))
97, 8syl 17 . . . . . 6 (𝜑 → (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶)))
105, 9mpbird 247 . . . . 5 (𝜑𝐴 ∈ {𝐴, 𝐵, 𝐶})
11 eqidd 2627 . . . . . . 7 (𝜑𝐵 = 𝐵)
12113mix2d 1235 . . . . . 6 (𝜑 → (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶))
136simp2d 1072 . . . . . . 7 (𝜑𝐵𝑌)
14 eltpg 4203 . . . . . . 7 (𝐵𝑌 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)))
1513, 14syl 17 . . . . . 6 (𝜑 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)))
1612, 15mpbird 247 . . . . 5 (𝜑𝐵 ∈ {𝐴, 𝐵, 𝐶})
17 f1fveq 6474 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 = 𝐵))
183, 10, 16, 17syl12anc 1321 . . . 4 (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 = 𝐵))
1918necon3bid 2840 . . 3 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐵) ↔ 𝐴𝐵))
202, 19mpbird 247 . 2 (𝜑 → (𝐹𝐴) ≠ (𝐹𝐵))
211simp2d 1072 . . 3 (𝜑𝐴𝐶)
226simp3d 1073 . . . . . 6 (𝜑𝐶𝑍)
23 tpid3g 4280 . . . . . 6 (𝐶𝑍𝐶 ∈ {𝐴, 𝐵, 𝐶})
2422, 23syl 17 . . . . 5 (𝜑𝐶 ∈ {𝐴, 𝐵, 𝐶})
25 f1fveq 6474 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
263, 10, 24, 25syl12anc 1321 . . . 4 (𝜑 → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
2726necon3bid 2840 . . 3 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐶) ↔ 𝐴𝐶))
2821, 27mpbird 247 . 2 (𝜑 → (𝐹𝐴) ≠ (𝐹𝐶))
291simp3d 1073 . . 3 (𝜑𝐵𝐶)
30 f1fveq 6474 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
313, 16, 24, 30syl12anc 1321 . . . 4 (𝜑 → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
3231necon3bid 2840 . . 3 (𝜑 → ((𝐹𝐵) ≠ (𝐹𝐶) ↔ 𝐵𝐶))
3329, 32mpbird 247 . 2 (𝜑 → (𝐹𝐵) ≠ (𝐹𝐶))
3420, 28, 333jca 1240 1 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐵) ∧ (𝐹𝐴) ≠ (𝐹𝐶) ∧ (𝐹𝐵) ≠ (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3o 1035  w3a 1036   = wceq 1480  wcel 1992  wne 2796  {ctp 4157  1-1wf1 5847  cfv 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fv 5858
This theorem is referenced by:  f1dom3el3dif  6481
  Copyright terms: Public domain W3C validator