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Theorem 2f1fvneq 7259
Description: If two one-to-one functions are applied on different arguments, also the values are different. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
Assertion
Ref Expression
2f1fvneq (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶) ∧ 𝐴𝐵) → (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → 𝑋𝑌))

Proof of Theorem 2f1fvneq
StepHypRef Expression
1 f1veqaeq 7256 . . . . 5 ((𝐹:𝐶1-1𝐷 ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) = (𝐹𝐵) → 𝐴 = 𝐵))
21adantll 713 . . . 4 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) = (𝐹𝐵) → 𝐴 = 𝐵))
32necon3ad 2954 . . 3 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → (𝐴𝐵 → ¬ (𝐹𝐴) = (𝐹𝐵)))
433impia 1118 . 2 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶) ∧ 𝐴𝐵) → ¬ (𝐹𝐴) = (𝐹𝐵))
5 simpll 766 . . . . . . 7 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → 𝐸:𝐷1-1𝑅)
6 f1f 6788 . . . . . . . . . 10 (𝐹:𝐶1-1𝐷𝐹:𝐶𝐷)
7 ffvelcdm 7084 . . . . . . . . . . . 12 ((𝐹:𝐶𝐷𝐴𝐶) → (𝐹𝐴) ∈ 𝐷)
8 ffvelcdm 7084 . . . . . . . . . . . 12 ((𝐹:𝐶𝐷𝐵𝐶) → (𝐹𝐵) ∈ 𝐷)
97, 8anim12dan 620 . . . . . . . . . . 11 ((𝐹:𝐶𝐷 ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷))
109ex 414 . . . . . . . . . 10 (𝐹:𝐶𝐷 → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
116, 10syl 17 . . . . . . . . 9 (𝐹:𝐶1-1𝐷 → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
1211adantl 483 . . . . . . . 8 ((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
1312imp 408 . . . . . . 7 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷))
14 f1veqaeq 7256 . . . . . . 7 ((𝐸:𝐷1-1𝑅 ∧ ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) → (𝐹𝐴) = (𝐹𝐵)))
155, 13, 14syl2anc 585 . . . . . 6 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) → (𝐹𝐴) = (𝐹𝐵)))
1615con3dimp 410 . . . . 5 ((((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) ∧ ¬ (𝐹𝐴) = (𝐹𝐵)) → ¬ (𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)))
17 eqeq12 2750 . . . . . . 7 (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) ↔ 𝑋 = 𝑌))
1817notbid 318 . . . . . 6 (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → (¬ (𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) ↔ ¬ 𝑋 = 𝑌))
19 neqne 2949 . . . . . 6 𝑋 = 𝑌𝑋𝑌)
2018, 19syl6bi 253 . . . . 5 (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → (¬ (𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) → 𝑋𝑌))
2116, 20syl5com 31 . . . 4 ((((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) ∧ ¬ (𝐹𝐴) = (𝐹𝐵)) → (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → 𝑋𝑌))
2221ex 414 . . 3 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → (¬ (𝐹𝐴) = (𝐹𝐵) → (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → 𝑋𝑌)))
23223adant3 1133 . 2 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶) ∧ 𝐴𝐵) → (¬ (𝐹𝐴) = (𝐹𝐵) → (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → 𝑋𝑌)))
244, 23mpd 15 1 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶) ∧ 𝐴𝐵) → (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → 𝑋𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wf 6540  1-1wf1 6541  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fv 6552
This theorem is referenced by:  usgr2pthlem  29020
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