Theorem cocan1 7292

Description: An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
cocan1	⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ 𝐻 = 𝐾))

Proof of Theorem cocan1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvco3 6978	. . . . . 6 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
2	1	3ad2antl2 1205	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
3		fvco3 6978	. . . . . 6 ⊢ ((𝐾:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐾)‘𝑥) = (𝐹‘(𝐾‘𝑥)))
4	3	3ad2antl3 1206	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐾)‘𝑥) = (𝐹‘(𝐾‘𝑥)))
5	2, 4	eqeq12d 2776	. . . 4 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → (((𝐹 ∘ 𝐻)‘𝑥) = ((𝐹 ∘ 𝐾)‘𝑥) ↔ (𝐹‘(𝐻‘𝑥)) = (𝐹‘(𝐾‘𝑥))))
6		simpl1 1210	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐵–1-1→𝐶)
7		ffvelcdm 7074	. . . . . 6 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
8	7	3ad2antl2 1205	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
9		ffvelcdm 7074	. . . . . 6 ⊢ ((𝐾:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐾‘𝑥) ∈ 𝐵)
10	9	3ad2antl3 1206	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐾‘𝑥) ∈ 𝐵)
11		f1fveq 7259	. . . . 5 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ ((𝐻‘𝑥) ∈ 𝐵 ∧ (𝐾‘𝑥) ∈ 𝐵)) → ((𝐹‘(𝐻‘𝑥)) = (𝐹‘(𝐾‘𝑥)) ↔ (𝐻‘𝑥) = (𝐾‘𝑥)))
12	6, 8, 10, 11	syl12anc 850	. . . 4 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘(𝐻‘𝑥)) = (𝐹‘(𝐾‘𝑥)) ↔ (𝐻‘𝑥) = (𝐾‘𝑥)))
13	5, 12	bitrd 282	. . 3 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → (((𝐹 ∘ 𝐻)‘𝑥) = ((𝐹 ∘ 𝐾)‘𝑥) ↔ (𝐻‘𝑥) = (𝐾‘𝑥)))
14	13	ralbidva 3183	. 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → (∀𝑥 ∈ 𝐴 ((𝐹 ∘ 𝐻)‘𝑥) = ((𝐹 ∘ 𝐾)‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐾‘𝑥)))
15		f1f 6771	. . . . . 6 ⊢ (𝐹:𝐵–1-1→𝐶 → 𝐹:𝐵⟶𝐶)
16	15	3ad2ant1 1151	. . . . 5 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐹:𝐵⟶𝐶)
17	16	ffnd 6703	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐹 Fn 𝐵)
18		simp2 1155	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐻:𝐴⟶𝐵)
19		fnfco 6740	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐻:𝐴⟶𝐵) → (𝐹 ∘ 𝐻) Fn 𝐴)
20	17, 18, 19	syl2anc 596	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → (𝐹 ∘ 𝐻) Fn 𝐴)
21		simp3 1156	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐾:𝐴⟶𝐵)
22		fnfco 6740	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐾:𝐴⟶𝐵) → (𝐹 ∘ 𝐾) Fn 𝐴)
23	17, 21, 22	syl2anc 596	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → (𝐹 ∘ 𝐾) Fn 𝐴)
24		eqfnfv 7022	. . 3 ⊢ (((𝐹 ∘ 𝐻) Fn 𝐴 ∧ (𝐹 ∘ 𝐾) Fn 𝐴) → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ ∀𝑥 ∈ 𝐴 ((𝐹 ∘ 𝐻)‘𝑥) = ((𝐹 ∘ 𝐾)‘𝑥)))
25	20, 23, 24	syl2anc 596	. 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ ∀𝑥 ∈ 𝐴 ((𝐹 ∘ 𝐻)‘𝑥) = ((𝐹 ∘ 𝐾)‘𝑥)))
26	18	ffnd 6703	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐻 Fn 𝐴)
27	21	ffnd 6703	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐾 Fn 𝐴)
28		eqfnfv 7022	. . 3 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐾 Fn 𝐴) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐾‘𝑥)))
29	26, 27, 28	syl2anc 596	. 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐾‘𝑥)))
30	14, 25, 29	3bitr4d 314	1 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ 𝐻 = 𝐾))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∘ ccom 5659   Fn wfn 6528  ⟶wf 6529  –1-1→wf1 6530  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fv 6541

This theorem is used by:  mapen  9139  mapfien  9378  hashfacen  14519  setcmon  18176  derangenlem  35750  subfacp1lem5  35763