Theorem cocan1 5427

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 5244	. . . . . 6 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
2	1	3ad2antl2 1067	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
3		fvco3 5244	. . . . . 6 ⊢ ((𝐾:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐾)‘𝑥) = (𝐹‘(𝐾‘𝑥)))
4	3	3ad2antl3 1068	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝐾)‘𝑥) = (𝐹‘(𝐾‘𝑥)))
5	2, 4	eqeq12d 2054	. . . 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 907	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐵–1-1→𝐶)
7		ffvelrn 5300	. . . . . 6 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
8	7	3ad2antl2 1067	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
9		ffvelrn 5300	. . . . . 6 ⊢ ((𝐾:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐾‘𝑥) ∈ 𝐵)
10	9	3ad2antl3 1068	. . . . 5 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐾‘𝑥) ∈ 𝐵)
11		f1fveq 5411	. . . . 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 1133	. . . 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 177	. . 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 2322	. 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 5092	. . . . . 6 ⊢ (𝐹:𝐵–1-1→𝐶 → 𝐹:𝐵⟶𝐶)
16	15	3ad2ant1 925	. . . . 5 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐹:𝐵⟶𝐶)
17		ffn 5046	. . . . 5 ⊢ (𝐹:𝐵⟶𝐶 → 𝐹 Fn 𝐵)
18	16, 17	syl 14	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐹 Fn 𝐵)
19		simp2 905	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐻:𝐴⟶𝐵)
20		fnfco 5065	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐻:𝐴⟶𝐵) → (𝐹 ∘ 𝐻) Fn 𝐴)
21	18, 19, 20	syl2anc 391	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → (𝐹 ∘ 𝐻) Fn 𝐴)
22		simp3 906	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐾:𝐴⟶𝐵)
23		fnfco 5065	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐾:𝐴⟶𝐵) → (𝐹 ∘ 𝐾) Fn 𝐴)
24	18, 22, 23	syl2anc 391	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → (𝐹 ∘ 𝐾) Fn 𝐴)
25		eqfnfv 5265	. . 3 ⊢ (((𝐹 ∘ 𝐻) Fn 𝐴 ∧ (𝐹 ∘ 𝐾) Fn 𝐴) → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ ∀𝑥 ∈ 𝐴 ((𝐹 ∘ 𝐻)‘𝑥) = ((𝐹 ∘ 𝐾)‘𝑥)))
26	21, 24, 25	syl2anc 391	. 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ ∀𝑥 ∈ 𝐴 ((𝐹 ∘ 𝐻)‘𝑥) = ((𝐹 ∘ 𝐾)‘𝑥)))
27		ffn 5046	. . . 4 ⊢ (𝐻:𝐴⟶𝐵 → 𝐻 Fn 𝐴)
28	19, 27	syl 14	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐻 Fn 𝐴)
29		ffn 5046	. . . 4 ⊢ (𝐾:𝐴⟶𝐵 → 𝐾 Fn 𝐴)
30	22, 29	syl 14	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → 𝐾 Fn 𝐴)
31		eqfnfv 5265	. . 3 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐾 Fn 𝐴) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐾‘𝑥)))
32	28, 30, 31	syl2anc 391	. 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐾‘𝑥)))
33	14, 26, 32	3bitr4d 209	1 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ 𝐻 = 𝐾))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∀wral 2306   ∘ ccom 4349   Fn wfn 4897  ⟶wf 4898  –1-1→wf1 4899  ‘cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fv 4910

This theorem is referenced by: (None)