MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2f1fvneq Structured version   Visualization version   GIF version

Theorem 2f1fvneq 7258
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 7255 . . . . 5 ((𝐹:𝐶1-1𝐷 ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) = (𝐹𝐵) → 𝐴 = 𝐵))
21adantll 712 . . . 4 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) = (𝐹𝐵) → 𝐴 = 𝐵))
32necon3ad 2953 . . 3 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → (𝐴𝐵 → ¬ (𝐹𝐴) = (𝐹𝐵)))
433impia 1117 . 2 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶) ∧ 𝐴𝐵) → ¬ (𝐹𝐴) = (𝐹𝐵))
5 simpll 765 . . . . . . 7 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → 𝐸:𝐷1-1𝑅)
6 f1f 6787 . . . . . . . . . 10 (𝐹:𝐶1-1𝐷𝐹:𝐶𝐷)
7 ffvelcdm 7083 . . . . . . . . . . . 12 ((𝐹:𝐶𝐷𝐴𝐶) → (𝐹𝐴) ∈ 𝐷)
8 ffvelcdm 7083 . . . . . . . . . . . 12 ((𝐹:𝐶𝐷𝐵𝐶) → (𝐹𝐵) ∈ 𝐷)
97, 8anim12dan 619 . . . . . . . . . . 11 ((𝐹:𝐶𝐷 ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷))
109ex 413 . . . . . . . . . 10 (𝐹:𝐶𝐷 → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
116, 10syl 17 . . . . . . . . 9 (𝐹:𝐶1-1𝐷 → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
1211adantl 482 . . . . . . . 8 ((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
1312imp 407 . . . . . . 7 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷))
14 f1veqaeq 7255 . . . . . . 7 ((𝐸:𝐷1-1𝑅 ∧ ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) → (𝐹𝐴) = (𝐹𝐵)))
155, 13, 14syl2anc 584 . . . . . 6 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) → (𝐹𝐴) = (𝐹𝐵)))
1615con3dimp 409 . . . . 5 ((((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) ∧ ¬ (𝐹𝐴) = (𝐹𝐵)) → ¬ (𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)))
17 eqeq12 2749 . . . . . . 7 (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) ↔ 𝑋 = 𝑌))
1817notbid 317 . . . . . 6 (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → (¬ (𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) ↔ ¬ 𝑋 = 𝑌))
19 neqne 2948 . . . . . 6 𝑋 = 𝑌𝑋𝑌)
2018, 19syl6bi 252 . . . . 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 413 . . 3 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → (¬ (𝐹𝐴) = (𝐹𝐵) → (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → 𝑋𝑌)))
23223adant3 1132 . 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 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wf 6539  1-1wf1 6540  cfv 6543
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551
This theorem is referenced by:  usgr2pthlem  29017
  Copyright terms: Public domain W3C validator