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Theorem 2f1fvneq 6738
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 6735 . . . . 5 ((𝐹:𝐶1-1𝐷 ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) = (𝐹𝐵) → 𝐴 = 𝐵))
21adantll 696 . . . 4 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) = (𝐹𝐵) → 𝐴 = 𝐵))
32necon3ad 2990 . . 3 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → (𝐴𝐵 → ¬ (𝐹𝐴) = (𝐹𝐵)))
433impia 1138 . 2 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶) ∧ 𝐴𝐵) → ¬ (𝐹𝐴) = (𝐹𝐵))
5 simpll 774 . . . . . . 7 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → 𝐸:𝐷1-1𝑅)
6 f1f 6313 . . . . . . . . . 10 (𝐹:𝐶1-1𝐷𝐹:𝐶𝐷)
7 ffvelrn 6576 . . . . . . . . . . . 12 ((𝐹:𝐶𝐷𝐴𝐶) → (𝐹𝐴) ∈ 𝐷)
8 ffvelrn 6576 . . . . . . . . . . . 12 ((𝐹:𝐶𝐷𝐵𝐶) → (𝐹𝐵) ∈ 𝐷)
97, 8anim12dan 607 . . . . . . . . . . 11 ((𝐹:𝐶𝐷 ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷))
109ex 399 . . . . . . . . . 10 (𝐹:𝐶𝐷 → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
116, 10syl 17 . . . . . . . . 9 (𝐹:𝐶1-1𝐷 → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
1211adantl 469 . . . . . . . 8 ((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
1312imp 395 . . . . . . 7 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷))
14 f1veqaeq 6735 . . . . . . 7 ((𝐸:𝐷1-1𝑅 ∧ ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) → (𝐹𝐴) = (𝐹𝐵)))
155, 13, 14syl2anc 575 . . . . . 6 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) → (𝐹𝐴) = (𝐹𝐵)))
1615con3dimp 397 . . . . 5 ((((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) ∧ ¬ (𝐹𝐴) = (𝐹𝐵)) → ¬ (𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)))
17 eqeq12 2818 . . . . . . 7 (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) ↔ 𝑋 = 𝑌))
1817notbid 309 . . . . . 6 (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → (¬ (𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) ↔ ¬ 𝑋 = 𝑌))
19 df-ne 2978 . . . . . . 7 (𝑋𝑌 ↔ ¬ 𝑋 = 𝑌)
2019biimpri 219 . . . . . 6 𝑋 = 𝑌𝑋𝑌)
2118, 20syl6bi 244 . . . . 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 399 . . 3 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → (¬ (𝐹𝐴) = (𝐹𝐵) → (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → 𝑋𝑌)))
24233adant3 1155 . 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 384  w3a 1100   = wceq 1637  wcel 2158  wne 2977  wf 6094  1-1wf1 6095  cfv 6098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pr 5093
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-rab 3104  df-v 3392  df-sbc 3631  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4627  df-br 4841  df-opab 4903  df-id 5216  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fv 6106
This theorem is referenced by:  usgr2pthlem  26883
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