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Theorem f1dom3fv3dif 6565
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 1093 . . 3 (𝜑𝐴𝐵)
3 f1dom3fv3dif.f . . . . 5 (𝜑𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅)
4 eqidd 2652 . . . . . . 7 (𝜑𝐴 = 𝐴)
543mix1d 1256 . . . . . 6 (𝜑 → (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶))
6 f1dom3fv3dif.v . . . . . . . 8 (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
76simp1d 1093 . . . . . . 7 (𝜑𝐴𝑋)
8 eltpg 4259 . . . . . . 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 2652 . . . . . . 7 (𝜑𝐵 = 𝐵)
12113mix2d 1257 . . . . . 6 (𝜑 → (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶))
136simp2d 1094 . . . . . . 7 (𝜑𝐵𝑌)
14 eltpg 4259 . . . . . . 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 6559 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 = 𝐵))
183, 10, 16, 17syl12anc 1364 . . . 4 (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 = 𝐵))
1918necon3bid 2867 . . 3 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐵) ↔ 𝐴𝐵))
202, 19mpbird 247 . 2 (𝜑 → (𝐹𝐴) ≠ (𝐹𝐵))
211simp2d 1094 . . 3 (𝜑𝐴𝐶)
226simp3d 1095 . . . . . 6 (𝜑𝐶𝑍)
23 tpid3g 4337 . . . . . 6 (𝐶𝑍𝐶 ∈ {𝐴, 𝐵, 𝐶})
2422, 23syl 17 . . . . 5 (𝜑𝐶 ∈ {𝐴, 𝐵, 𝐶})
25 f1fveq 6559 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
263, 10, 24, 25syl12anc 1364 . . . 4 (𝜑 → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
2726necon3bid 2867 . . 3 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐶) ↔ 𝐴𝐶))
2821, 27mpbird 247 . 2 (𝜑 → (𝐹𝐴) ≠ (𝐹𝐶))
291simp3d 1095 . . 3 (𝜑𝐵𝐶)
30 f1fveq 6559 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
313, 16, 24, 30syl12anc 1364 . . . 4 (𝜑 → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
3231necon3bid 2867 . . 3 (𝜑 → ((𝐹𝐵) ≠ (𝐹𝐶) ↔ 𝐵𝐶))
3329, 32mpbird 247 . 2 (𝜑 → (𝐹𝐵) ≠ (𝐹𝐶))
3420, 28, 333jca 1261 1 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐵) ∧ (𝐹𝐴) ≠ (𝐹𝐶) ∧ (𝐹𝐵) ≠ (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3o 1053  w3a 1054   = wceq 1523  wcel 2030  wne 2823  {ctp 4214  1-1wf1 5923  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fv 5934
This theorem is referenced by:  f1dom3el3dif  6566
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