Theorem cocan1 7237

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 6940	. . . . . 6 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
2	1	3ad2antl2 1186	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
3		fvco3 6940	. . . . . 6 ⊢ ((𝐾:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐾)‘𝑥) = (𝐹‘(𝐾‘𝑥)))
4	3	3ad2antl3 1187	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐾)‘𝑥) = (𝐹‘(𝐾‘𝑥)))
5	2, 4	eqeq12d 2752	. . . 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 1191	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐵–1-1→𝐶)
7		ffvelcdm 7032	. . . . . 6 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
8	7	3ad2antl2 1186	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
9		ffvelcdm 7032	. . . . . 6 ⊢ ((𝐾:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐾‘𝑥) ∈ 𝐵)
10	9	3ad2antl3 1187	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐾‘𝑥) ∈ 𝐵)
11		f1fveq 7209	. . . . 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 835	. . . 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 3172	. 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 6738	. . . . . 6 ⊢ (𝐹:𝐵–1-1→𝐶 → 𝐹:𝐵⟶𝐶)
16	15	3ad2ant1 1133	. . . . 5 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐹:𝐵⟶𝐶)
17	16	ffnd 6669	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐹 Fn 𝐵)
18		simp2 1137	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐻:𝐴⟶𝐵)
19		fnfco 6707	. . . 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 1138	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐾:𝐴⟶𝐵)
22		fnfco 6707	. . . 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 6982	. . 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 6669	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐻 Fn 𝐴)
27	21	ffnd 6669	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐾 Fn 𝐴)
28		eqfnfv 6982	. . 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 310	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 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064   ∘ ccom 5637   Fn wfn 6491  ⟶wf 6492  –1-1→wf1 6493  ‘cfv 6496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fv 6504

This theorem is referenced by:  mapen  9085  mapfien  9344  hashfacen  14351  hashfacenOLD  14352  setcmon  17973  derangenlem  33765  subfacp1lem5  33778