Theorem cocan1 7163

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 6867	. . . . . 6 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
2	1	3ad2antl2 1185	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
3		fvco3 6867	. . . . . 6 ⊢ ((𝐾:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐾)‘𝑥) = (𝐹‘(𝐾‘𝑥)))
4	3	3ad2antl3 1186	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐾)‘𝑥) = (𝐹‘(𝐾‘𝑥)))
5	2, 4	eqeq12d 2754	. . . 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 1190	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐵–1-1→𝐶)
7		ffvelrn 6959	. . . . . 6 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
8	7	3ad2antl2 1185	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
9		ffvelrn 6959	. . . . . 6 ⊢ ((𝐾:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐾‘𝑥) ∈ 𝐵)
10	9	3ad2antl3 1186	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐾‘𝑥) ∈ 𝐵)
11		f1fveq 7135	. . . . 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 834	. . . 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 278	. . 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 3111	. 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 6670	. . . . . 6 ⊢ (𝐹:𝐵–1-1→𝐶 → 𝐹:𝐵⟶𝐶)
16	15	3ad2ant1 1132	. . . . 5 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐹:𝐵⟶𝐶)
17	16	ffnd 6601	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐹 Fn 𝐵)
18		simp2 1136	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐻:𝐴⟶𝐵)
19		fnfco 6639	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐻:𝐴⟶𝐵) → (𝐹 ∘ 𝐻) Fn 𝐴)
20	17, 18, 19	syl2anc 584	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → (𝐹 ∘ 𝐻) Fn 𝐴)
21		simp3 1137	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐾:𝐴⟶𝐵)
22		fnfco 6639	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐾:𝐴⟶𝐵) → (𝐹 ∘ 𝐾) Fn 𝐴)
23	17, 21, 22	syl2anc 584	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → (𝐹 ∘ 𝐾) Fn 𝐴)
24		eqfnfv 6909	. . 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 584	. 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 6601	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐻 Fn 𝐴)
27	21	ffnd 6601	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐾 Fn 𝐴)
28		eqfnfv 6909	. . 3 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐾 Fn 𝐴) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐾‘𝑥)))
29	26, 27, 28	syl2anc 584	. 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 311	1 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ 𝐻 = 𝐾))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  –1-1→wf1 6430  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fv 6441

This theorem is referenced by:  mapen  8928  mapfien  9167  hashfacen  14166  hashfacenOLD  14167  setcmon  17802  derangenlem  33133  subfacp1lem5  33146