Theorem 2f1fvneq 7280

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:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → 𝑋 ≠ 𝑌))

Proof of Theorem 2f1fvneq

Step	Hyp	Ref	Expression
1		f1veqaeq 7277	. . . . 5 ⊢ ((𝐹:𝐶–1-1→𝐷 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐴) = (𝐹‘𝐵) → 𝐴 = 𝐵))
2	1	adantll 714	. . . 4 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐴) = (𝐹‘𝐵) → 𝐴 = 𝐵))
3	2	necon3ad 2951	. . 3 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (𝐴 ≠ 𝐵 → ¬ (𝐹‘𝐴) = (𝐹‘𝐵)))
4	3	3impia 1116	. 2 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → ¬ (𝐹‘𝐴) = (𝐹‘𝐵))
5		simpll 767	. . . . . . 7 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → 𝐸:𝐷–1-1→𝑅)
6		f1f 6805	. . . . . . . . . 10 ⊢ (𝐹:𝐶–1-1→𝐷 → 𝐹:𝐶⟶𝐷)
7		ffvelcdm 7101	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐶⟶𝐷 ∧ 𝐴 ∈ 𝐶) → (𝐹‘𝐴) ∈ 𝐷)
8		ffvelcdm 7101	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐶⟶𝐷 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝐵) ∈ 𝐷)
9	7, 8	anim12dan 619	. . . . . . . . . . 11 ⊢ ((𝐹:𝐶⟶𝐷 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐴) ∈ 𝐷 ∧ (𝐹‘𝐵) ∈ 𝐷))
10	9	ex 412	. . . . . . . . . 10 ⊢ (𝐹:𝐶⟶𝐷 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐹‘𝐴) ∈ 𝐷 ∧ (𝐹‘𝐵) ∈ 𝐷)))
11	6, 10	syl 17	. . . . . . . . 9 ⊢ (𝐹:𝐶–1-1→𝐷 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐹‘𝐴) ∈ 𝐷 ∧ (𝐹‘𝐵) ∈ 𝐷)))
12	11	adantl 481	. . . . . . . 8 ⊢ ((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐹‘𝐴) ∈ 𝐷 ∧ (𝐹‘𝐵) ∈ 𝐷)))
13	12	imp 406	. . . . . . 7 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐴) ∈ 𝐷 ∧ (𝐹‘𝐵) ∈ 𝐷))
14		f1veqaeq 7277	. . . . . . 7 ⊢ ((𝐸:𝐷–1-1→𝑅 ∧ ((𝐹‘𝐴) ∈ 𝐷 ∧ (𝐹‘𝐵) ∈ 𝐷)) → ((𝐸‘(𝐹‘𝐴)) = (𝐸‘(𝐹‘𝐵)) → (𝐹‘𝐴) = (𝐹‘𝐵)))
15	5, 13, 14	syl2anc 584	. . . . . 6 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐸‘(𝐹‘𝐴)) = (𝐸‘(𝐹‘𝐵)) → (𝐹‘𝐴) = (𝐹‘𝐵)))
16	15	con3dimp 408	. . . . 5 ⊢ ((((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) ∧ ¬ (𝐹‘𝐴) = (𝐹‘𝐵)) → ¬ (𝐸‘(𝐹‘𝐴)) = (𝐸‘(𝐹‘𝐵)))
17		eqeq12 2752	. . . . . . 7 ⊢ (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → ((𝐸‘(𝐹‘𝐴)) = (𝐸‘(𝐹‘𝐵)) ↔ 𝑋 = 𝑌))
18	17	notbid 318	. . . . . 6 ⊢ (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → (¬ (𝐸‘(𝐹‘𝐴)) = (𝐸‘(𝐹‘𝐵)) ↔ ¬ 𝑋 = 𝑌))
19		neqne 2946	. . . . . 6 ⊢ (¬ 𝑋 = 𝑌 → 𝑋 ≠ 𝑌)
20	18, 19	biimtrdi 253	. . . . 5 ⊢ (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → (¬ (𝐸‘(𝐹‘𝐴)) = (𝐸‘(𝐹‘𝐵)) → 𝑋 ≠ 𝑌))
21	16, 20	syl5com 31	. . . 4 ⊢ ((((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) ∧ ¬ (𝐹‘𝐴) = (𝐹‘𝐵)) → (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → 𝑋 ≠ 𝑌))
22	21	ex 412	. . 3 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (¬ (𝐹‘𝐴) = (𝐹‘𝐵) → (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → 𝑋 ≠ 𝑌)))
23	22	3adant3 1131	. 2 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → (¬ (𝐹‘𝐴) = (𝐹‘𝐵) → (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → 𝑋 ≠ 𝑌)))
24	4, 23	mpd 15	1 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → 𝑋 ≠ 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ⟶wf 6559  –1-1→wf1 6560  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571

This theorem is referenced by:  usgr2pthlem  29796