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Theorem 2f1fvneq 6557
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 6554 . . . . 5 ((𝐹:𝐶1-1𝐷 ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) = (𝐹𝐵) → 𝐴 = 𝐵))
21adantll 750 . . . 4 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) = (𝐹𝐵) → 𝐴 = 𝐵))
32necon3ad 2836 . . 3 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → (𝐴𝐵 → ¬ (𝐹𝐴) = (𝐹𝐵)))
433impia 1280 . 2 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶) ∧ 𝐴𝐵) → ¬ (𝐹𝐴) = (𝐹𝐵))
5 simpll 805 . . . . . . 7 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → 𝐸:𝐷1-1𝑅)
6 f1f 6139 . . . . . . . . . 10 (𝐹:𝐶1-1𝐷𝐹:𝐶𝐷)
7 ffvelrn 6397 . . . . . . . . . . . 12 ((𝐹:𝐶𝐷𝐴𝐶) → (𝐹𝐴) ∈ 𝐷)
8 ffvelrn 6397 . . . . . . . . . . . 12 ((𝐹:𝐶𝐷𝐵𝐶) → (𝐹𝐵) ∈ 𝐷)
97, 8anim12dan 900 . . . . . . . . . . 11 ((𝐹:𝐶𝐷 ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷))
109ex 449 . . . . . . . . . 10 (𝐹:𝐶𝐷 → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
116, 10syl 17 . . . . . . . . 9 (𝐹:𝐶1-1𝐷 → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
1211adantl 481 . . . . . . . 8 ((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) → ((𝐴𝐶𝐵𝐶) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)))
1312imp 444 . . . . . . 7 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷))
14 f1veqaeq 6554 . . . . . . 7 ((𝐸:𝐷1-1𝑅 ∧ ((𝐹𝐴) ∈ 𝐷 ∧ (𝐹𝐵) ∈ 𝐷)) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) → (𝐹𝐴) = (𝐹𝐵)))
155, 13, 14syl2anc 694 . . . . . 6 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) → (𝐹𝐴) = (𝐹𝐵)))
1615con3dimp 456 . . . . 5 ((((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) ∧ ¬ (𝐹𝐴) = (𝐹𝐵)) → ¬ (𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)))
17 eqeq12 2664 . . . . . . 7 (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → ((𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) ↔ 𝑋 = 𝑌))
1817notbid 307 . . . . . 6 (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → (¬ (𝐸‘(𝐹𝐴)) = (𝐸‘(𝐹𝐵)) ↔ ¬ 𝑋 = 𝑌))
19 df-ne 2824 . . . . . . 7 (𝑋𝑌 ↔ ¬ 𝑋 = 𝑌)
2019biimpri 218 . . . . . 6 𝑋 = 𝑌𝑋𝑌)
2118, 20syl6bi 243 . . . . 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 449 . . 3 (((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶)) → (¬ (𝐹𝐴) = (𝐹𝐵) → (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → 𝑋𝑌)))
24233adant3 1101 . 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 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wf 5922  1-1wf1 5923  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fv 5934
This theorem is referenced by:  usgr2pthlem  26715
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