Theorem 2f1fvneq 7029

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 7026	. . . . 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 2947	. . 3 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (𝐴 ≠ 𝐵 → ¬ (𝐹‘𝐴) = (𝐹‘𝐵)))
4	3	3impia 1118	. 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 6574	. . . . . . . . . 10 ⊢ (𝐹:𝐶–1-1→𝐷 → 𝐹:𝐶⟶𝐷)
7		ffvelrn 6859	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐶⟶𝐷 ∧ 𝐴 ∈ 𝐶) → (𝐹‘𝐴) ∈ 𝐷)
8		ffvelrn 6859	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐶⟶𝐷 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝐵) ∈ 𝐷)
9	7, 8	anim12dan 622	. . . . . . . . . . 11 ⊢ ((𝐹:𝐶⟶𝐷 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐴) ∈ 𝐷 ∧ (𝐹‘𝐵) ∈ 𝐷))
10	9	ex 416	. . . . . . . . . 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 485	. . . . . . . 8 ⊢ ((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐹‘𝐴) ∈ 𝐷 ∧ (𝐹‘𝐵) ∈ 𝐷)))
13	12	imp 410	. . . . . . 7 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐴) ∈ 𝐷 ∧ (𝐹‘𝐵) ∈ 𝐷))
14		f1veqaeq 7026	. . . . . . 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 587	. . . . . 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 412	. . . . 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 321	. . . . . 6 ⊢ (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → (¬ (𝐸‘(𝐹‘𝐴)) = (𝐸‘(𝐹‘𝐵)) ↔ ¬ 𝑋 = 𝑌))
19		neqne 2942	. . . . . 6 ⊢ (¬ 𝑋 = 𝑌 → 𝑋 ≠ 𝑌)
20	18, 19	syl6bi 256	. . . . 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 416	. . 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 1133	. 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 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114   ≠ wne 2934  ⟶wf 6335  –1-1→wf1 6336  ‘cfv 6339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fv 6347

This theorem is referenced by:  usgr2pthlem  27704