Theorem 2f1fvneq 7012

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 7009	. . . . 5 ⊢ ((𝐹:𝐶–1-1→𝐷 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐴) = (𝐹‘𝐵) → 𝐴 = 𝐵))
2	1	adantll 712	. . . 4 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐴) = (𝐹‘𝐵) → 𝐴 = 𝐵))
3	2	necon3ad 3029	. . 3 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (𝐴 ≠ 𝐵 → ¬ (𝐹‘𝐴) = (𝐹‘𝐵)))
4	3	3impia 1113	. 2 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → ¬ (𝐹‘𝐴) = (𝐹‘𝐵))
5		simpll 765	. . . . . . 7 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → 𝐸:𝐷–1-1→𝑅)
6		f1f 6570	. . . . . . . . . 10 ⊢ (𝐹:𝐶–1-1→𝐷 → 𝐹:𝐶⟶𝐷)
7		ffvelrn 6844	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐶⟶𝐷 ∧ 𝐴 ∈ 𝐶) → (𝐹‘𝐴) ∈ 𝐷)
8		ffvelrn 6844	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐶⟶𝐷 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝐵) ∈ 𝐷)
9	7, 8	anim12dan 620	. . . . . . . . . . 11 ⊢ ((𝐹:𝐶⟶𝐷 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐴) ∈ 𝐷 ∧ (𝐹‘𝐵) ∈ 𝐷))
10	9	ex 415	. . . . . . . . . 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 484	. . . . . . . 8 ⊢ ((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐹‘𝐴) ∈ 𝐷 ∧ (𝐹‘𝐵) ∈ 𝐷)))
13	12	imp 409	. . . . . . 7 ⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐴) ∈ 𝐷 ∧ (𝐹‘𝐵) ∈ 𝐷))
14		f1veqaeq 7009	. . . . . . 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 586	. . . . . 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 411	. . . . 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 2835	. . . . . . 7 ⊢ (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → ((𝐸‘(𝐹‘𝐴)) = (𝐸‘(𝐹‘𝐵)) ↔ 𝑋 = 𝑌))
18	17	notbid 320	. . . . . 6 ⊢ (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → (¬ (𝐸‘(𝐹‘𝐴)) = (𝐸‘(𝐹‘𝐵)) ↔ ¬ 𝑋 = 𝑌))
19		neqne 3024	. . . . . 6 ⊢ (¬ 𝑋 = 𝑌 → 𝑋 ≠ 𝑌)
20	18, 19	syl6bi 255	. . . . 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 415	. . 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 1128	. 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 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ⟶wf 6346  –1-1→wf1 6347  ‘cfv 6350

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fv 6358

This theorem is referenced by:  usgr2pthlem  27538