Theorem cocan1 7247

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 6941	. . . . . 6 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
2	1	3ad2antl2 1188	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
3		fvco3 6941	. . . . . 6 ⊢ ((𝐾:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐾)‘𝑥) = (𝐹‘(𝐾‘𝑥)))
4	3	3ad2antl3 1189	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐾)‘𝑥) = (𝐹‘(𝐾‘𝑥)))
5	2, 4	eqeq12d 2753	. . . 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 1193	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐵–1-1→𝐶)
7		ffvelcdm 7035	. . . . . 6 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
8	7	3ad2antl2 1188	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
9		ffvelcdm 7035	. . . . . 6 ⊢ ((𝐾:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐾‘𝑥) ∈ 𝐵)
10	9	3ad2antl3 1189	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐾‘𝑥) ∈ 𝐵)
11		f1fveq 7218	. . . . 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 837	. . . 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 279	. . 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 3159	. 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 1134	. . . . 5 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐹:𝐵⟶𝐶)
17	16	ffnd 6671	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐹 Fn 𝐵)
18		simp2 1138	. . . 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 585	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → (𝐹 ∘ 𝐻) Fn 𝐴)
21		simp3 1139	. . . 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 585	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → (𝐹 ∘ 𝐾) Fn 𝐴)
24		eqfnfv 6985	. . 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 585	. 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 6671	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐻 Fn 𝐴)
27	21	ffnd 6671	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐾 Fn 𝐴)
28		eqfnfv 6985	. . 3 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐾 Fn 𝐴) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐾‘𝑥)))
29	26, 27, 28	syl2anc 585	. 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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∘ ccom 5636   Fn wfn 6495  ⟶wf 6496  –1-1→wf1 6497  ‘cfv 6500

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fv 6508

This theorem is referenced by:  mapen  9081  mapfien  9323  hashfacen  14389  setcmon  18023  derangenlem  35387  subfacp1lem5  35400