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Theorem f1dom3fv3dif 7017
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 1134 . . 3 (𝜑𝐴𝐵)
3 f1dom3fv3dif.f . . . . 5 (𝜑𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅)
4 eqidd 2819 . . . . . . 7 (𝜑𝐴 = 𝐴)
543mix1d 1328 . . . . . 6 (𝜑 → (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶))
6 f1dom3fv3dif.v . . . . . . . 8 (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
76simp1d 1134 . . . . . . 7 (𝜑𝐴𝑋)
8 eltpg 4615 . . . . . . 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 258 . . . . 5 (𝜑𝐴 ∈ {𝐴, 𝐵, 𝐶})
11 eqidd 2819 . . . . . . 7 (𝜑𝐵 = 𝐵)
12113mix2d 1329 . . . . . 6 (𝜑 → (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶))
136simp2d 1135 . . . . . . 7 (𝜑𝐵𝑌)
14 eltpg 4615 . . . . . . 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 258 . . . . 5 (𝜑𝐵 ∈ {𝐴, 𝐵, 𝐶})
17 f1fveq 7011 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 = 𝐵))
183, 10, 16, 17syl12anc 832 . . . 4 (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 = 𝐵))
1918necon3bid 3057 . . 3 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐵) ↔ 𝐴𝐵))
202, 19mpbird 258 . 2 (𝜑 → (𝐹𝐴) ≠ (𝐹𝐵))
211simp2d 1135 . . 3 (𝜑𝐴𝐶)
226simp3d 1136 . . . . . 6 (𝜑𝐶𝑍)
23 tpid3g 4700 . . . . . 6 (𝐶𝑍𝐶 ∈ {𝐴, 𝐵, 𝐶})
2422, 23syl 17 . . . . 5 (𝜑𝐶 ∈ {𝐴, 𝐵, 𝐶})
25 f1fveq 7011 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
263, 10, 24, 25syl12anc 832 . . . 4 (𝜑 → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
2726necon3bid 3057 . . 3 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐶) ↔ 𝐴𝐶))
2821, 27mpbird 258 . 2 (𝜑 → (𝐹𝐴) ≠ (𝐹𝐶))
291simp3d 1136 . . 3 (𝜑𝐵𝐶)
30 f1fveq 7011 . . . . 5 ((𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅 ∧ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶})) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
313, 16, 24, 30syl12anc 832 . . . 4 (𝜑 → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
3231necon3bid 3057 . . 3 (𝜑 → ((𝐹𝐵) ≠ (𝐹𝐶) ↔ 𝐵𝐶))
3329, 32mpbird 258 . 2 (𝜑 → (𝐹𝐵) ≠ (𝐹𝐶))
3420, 28, 333jca 1120 1 (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐵) ∧ (𝐹𝐴) ≠ (𝐹𝐶) ∧ (𝐹𝐵) ≠ (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3o 1078  w3a 1079   = wceq 1528  wcel 2105  wne 3013  {ctp 4561  1-1wf1 6345  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fv 6356
This theorem is referenced by:  f1dom3el3dif  7018
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