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

Theorem 2f1fvneq 6772
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 6769 . . . . 5 ((𝐹:𝐶1-1𝐷 ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) = (𝐹𝐵) → 𝐴 = 𝐵))
21adantll 707 . . . 4 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) = (𝐹𝐵) → 𝐴 = 𝐵))
32necon3ad 3012 . . 3 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → (𝐴𝐵 → ¬ (𝐹𝐴) = (𝐹𝐵)))
433impia 1151 . 2 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶) ∧ 𝐴𝐵) → ¬ (𝐹𝐴) = (𝐹𝐵))
5 simpll 785 . . . . . . 7 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → 𝐸:𝐷1-1𝑅)
6 f1f 6338 . . . . . . . . . 10 (𝐹:𝐶1-1𝐷𝐹:𝐶𝐷)
7 ffvelrn 6606 . . . . . . . . . . . 12 ((𝐹:𝐶𝐷𝐴𝐶) → (𝐹𝐴) ∈ 𝐷)
8 ffvelrn 6606 . . . . . . . . . . . 12 ((𝐹:𝐶𝐷𝐵𝐶) → (𝐹𝐵) ∈ 𝐷)
97, 8anim12dan 614 . . . . . . . . . . 11 ((𝐹:𝐶𝐷 ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷))
109ex 403 . . . . . . . . . 10 (𝐹:𝐶𝐷 → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
116, 10syl 17 . . . . . . . . 9 (𝐹:𝐶1-1𝐷 → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
1211adantl 475 . . . . . . . 8 ((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
1312imp 397 . . . . . . 7 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷))
14 f1veqaeq 6769 . . . . . . 7 ((𝐸:𝐷1-1𝑅 ∧ ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) → (𝐹𝐴) = (𝐹𝐵)))
155, 13, 14syl2anc 581 . . . . . 6 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) → (𝐹𝐴) = (𝐹𝐵)))
1615con3dimp 399 . . . . 5 ((((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) ∧ ¬ (𝐹𝐴) = (𝐹𝐵)) → ¬ (𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)))
17 eqeq12 2838 . . . . . . 7 (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) ↔ 𝑋 = 𝑌))
1817notbid 310 . . . . . 6 (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → (¬ (𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) ↔ ¬ 𝑋 = 𝑌))
19 df-ne 3000 . . . . . . 7 (𝑋𝑌 ↔ ¬ 𝑋 = 𝑌)
2019biimpri 220 . . . . . 6 𝑋 = 𝑌𝑋𝑌)
2118, 20syl6bi 245 . . . . 5 (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → (¬ (𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) → 𝑋𝑌))
2216, 21syl5com 31 . . . 4 ((((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) ∧ ¬ (𝐹𝐴) = (𝐹𝐵)) → (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → 𝑋𝑌))
2322ex 403 . . 3 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → (¬ (𝐹𝐴) = (𝐹𝐵) → (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → 𝑋𝑌)))
24233adant3 1168 . 2 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶) ∧ 𝐴𝐵) → (¬ (𝐹𝐴) = (𝐹𝐵) → (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → 𝑋𝑌)))
254, 24mpd 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 386  w3a 1113   = wceq 1658  wcel 2166  wne 2999  wf 6119  1-1wf1 6120  cfv 6123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fv 6131
This theorem is referenced by:  usgr2pthlem  27065
  Copyright terms: Public domain W3C validator