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Theorem f1dom3fv3dif 6797
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 1133 . . 3 (𝜑𝐴𝐵)
3 f1dom3fv3dif.f . . . . 5 (𝜑𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅)
4 eqidd 2779 . . . . . . 7 (𝜑𝐴 = 𝐴)
543mix1d 1392 . . . . . 6 (𝜑 → (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶))
6 f1dom3fv3dif.v . . . . . . . 8 (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
76simp1d 1133 . . . . . . 7 (𝜑𝐴𝑋)
8 eltpg 4454 . . . . . . 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 249 . . . . 5 (𝜑𝐴 ∈ {𝐴, 𝐵, 𝐶})
11 eqidd 2779 . . . . . . 7 (𝜑𝐵 = 𝐵)
12113mix2d 1393 . . . . . 6 (𝜑 → (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶))
136simp2d 1134 . . . . . . 7 (𝜑𝐵𝑌)
14 eltpg 4454 . . . . . . 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 249 . . . . 5 (𝜑𝐵 ∈ {𝐴, 𝐵, 𝐶})
17 f1fveq 6791 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 = 𝐵))
183, 10, 16, 17syl12anc 827 . . . 4 (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 = 𝐵))
1918necon3bid 3013 . . 3 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐵) ↔ 𝐴𝐵))
202, 19mpbird 249 . 2 (𝜑 → (𝐹𝐴) ≠ (𝐹𝐵))
211simp2d 1134 . . 3 (𝜑𝐴𝐶)
226simp3d 1135 . . . . . 6 (𝜑𝐶𝑍)
23 tpid3g 4539 . . . . . 6 (𝐶𝑍𝐶 ∈ {𝐴, 𝐵, 𝐶})
2422, 23syl 17 . . . . 5 (𝜑𝐶 ∈ {𝐴, 𝐵, 𝐶})
25 f1fveq 6791 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
263, 10, 24, 25syl12anc 827 . . . 4 (𝜑 → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
2726necon3bid 3013 . . 3 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐶) ↔ 𝐴𝐶))
2821, 27mpbird 249 . 2 (𝜑 → (𝐹𝐴) ≠ (𝐹𝐶))
291simp3d 1135 . . 3 (𝜑𝐵𝐶)
30 f1fveq 6791 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
313, 16, 24, 30syl12anc 827 . . . 4 (𝜑 → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
3231necon3bid 3013 . . 3 (𝜑 → ((𝐹𝐵) ≠ (𝐹𝐶) ↔ 𝐵𝐶))
3329, 32mpbird 249 . 2 (𝜑 → (𝐹𝐵) ≠ (𝐹𝐶))
3420, 28, 333jca 1119 1 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐵) ∧ (𝐹𝐴) ≠ (𝐹𝐶) ∧ (𝐹𝐵) ≠ (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3o 1070  w3a 1071   = wceq 1601  wcel 2107  wne 2969  {ctp 4402  1-1wf1 6132  cfv 6135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fv 6143
This theorem is referenced by:  f1dom3el3dif  6798
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