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Theorem f1dom3fv3dif 7008
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 1139 . . 3 (𝜑𝐴𝐵)
3 f1dom3fv3dif.f . . . . 5 (𝜑𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅)
4 eqidd 2802 . . . . . . 7 (𝜑𝐴 = 𝐴)
543mix1d 1333 . . . . . 6 (𝜑 → (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶))
6 f1dom3fv3dif.v . . . . . . . 8 (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
76simp1d 1139 . . . . . . 7 (𝜑𝐴𝑋)
8 eltpg 4586 . . . . . . 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 260 . . . . 5 (𝜑𝐴 ∈ {𝐴, 𝐵, 𝐶})
11 eqidd 2802 . . . . . . 7 (𝜑𝐵 = 𝐵)
12113mix2d 1334 . . . . . 6 (𝜑 → (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶))
136simp2d 1140 . . . . . . 7 (𝜑𝐵𝑌)
14 eltpg 4586 . . . . . . 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 260 . . . . 5 (𝜑𝐵 ∈ {𝐴, 𝐵, 𝐶})
17 f1fveq 7002 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 = 𝐵))
183, 10, 16, 17syl12anc 835 . . . 4 (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 = 𝐵))
1918necon3bid 3034 . . 3 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐵) ↔ 𝐴𝐵))
202, 19mpbird 260 . 2 (𝜑 → (𝐹𝐴) ≠ (𝐹𝐵))
211simp2d 1140 . . 3 (𝜑𝐴𝐶)
226simp3d 1141 . . . . . 6 (𝜑𝐶𝑍)
23 tpid3g 4671 . . . . . 6 (𝐶𝑍𝐶 ∈ {𝐴, 𝐵, 𝐶})
2422, 23syl 17 . . . . 5 (𝜑𝐶 ∈ {𝐴, 𝐵, 𝐶})
25 f1fveq 7002 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
263, 10, 24, 25syl12anc 835 . . . 4 (𝜑 → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
2726necon3bid 3034 . . 3 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐶) ↔ 𝐴𝐶))
2821, 27mpbird 260 . 2 (𝜑 → (𝐹𝐴) ≠ (𝐹𝐶))
291simp3d 1141 . . 3 (𝜑𝐵𝐶)
30 f1fveq 7002 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
313, 16, 24, 30syl12anc 835 . . . 4 (𝜑 → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
3231necon3bid 3034 . . 3 (𝜑 → ((𝐹𝐵) ≠ (𝐹𝐶) ↔ 𝐵𝐶))
3329, 32mpbird 260 . 2 (𝜑 → (𝐹𝐵) ≠ (𝐹𝐶))
3420, 28, 333jca 1125 1 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐵) ∧ (𝐹𝐴) ≠ (𝐹𝐶) ∧ (𝐹𝐵) ≠ (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3o 1083  w3a 1084   = wceq 1538  wcel 2112  wne 2990  {ctp 4532  1-1wf1 6325  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fv 6336
This theorem is referenced by:  f1dom3el3dif  7009
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