Theorem cocan2 5792

Description: A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
cocan2	⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ 𝐻 = 𝐾))

Proof of Theorem cocan2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fof 5440	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	3ad2ant1 1018	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐹:𝐴⟶𝐵)
3		fvco3 5590	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
4	2, 3	sylan 283	. . . . 5 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
5		fvco3 5590	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑦) = (𝐾‘(𝐹‘𝑦)))
6	2, 5	sylan 283	. . . . 5 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑦) = (𝐾‘(𝐹‘𝑦)))
7	4, 6	eqeq12d 2192	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) ∧ 𝑦 ∈ 𝐴) → (((𝐻 ∘ 𝐹)‘𝑦) = ((𝐾 ∘ 𝐹)‘𝑦) ↔ (𝐻‘(𝐹‘𝑦)) = (𝐾‘(𝐹‘𝑦))))
8	7	ralbidva 2473	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (∀𝑦 ∈ 𝐴 ((𝐻 ∘ 𝐹)‘𝑦) = ((𝐾 ∘ 𝐹)‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝐻‘(𝐹‘𝑦)) = (𝐾‘(𝐹‘𝑦))))
9		fveq2 5517	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐻‘(𝐹‘𝑦)) = (𝐻‘𝑥))
10		fveq2 5517	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐾‘(𝐹‘𝑦)) = (𝐾‘𝑥))
11	9, 10	eqeq12d 2192	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 → ((𝐻‘(𝐹‘𝑦)) = (𝐾‘(𝐹‘𝑦)) ↔ (𝐻‘𝑥) = (𝐾‘𝑥)))
12	11	cbvfo 5789	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ 𝐴 (𝐻‘(𝐹‘𝑦)) = (𝐾‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
13	12	3ad2ant1 1018	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (∀𝑦 ∈ 𝐴 (𝐻‘(𝐹‘𝑦)) = (𝐾‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
14	8, 13	bitrd 188	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (∀𝑦 ∈ 𝐴 ((𝐻 ∘ 𝐹)‘𝑦) = ((𝐾 ∘ 𝐹)‘𝑦) ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
15		simp2 998	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐻 Fn 𝐵)
16		fnfco 5392	. . . 4 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐻 ∘ 𝐹) Fn 𝐴)
17	15, 2, 16	syl2anc 411	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 ∘ 𝐹) Fn 𝐴)
18		simp3 999	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
19		fnfco 5392	. . . 4 ⊢ ((𝐾 Fn 𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐾 ∘ 𝐹) Fn 𝐴)
20	18, 2, 19	syl2anc 411	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐾 ∘ 𝐹) Fn 𝐴)
21		eqfnfv 5616	. . 3 ⊢ (((𝐻 ∘ 𝐹) Fn 𝐴 ∧ (𝐾 ∘ 𝐹) Fn 𝐴) → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ ∀𝑦 ∈ 𝐴 ((𝐻 ∘ 𝐹)‘𝑦) = ((𝐾 ∘ 𝐹)‘𝑦)))
22	17, 20, 21	syl2anc 411	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ ∀𝑦 ∈ 𝐴 ((𝐻 ∘ 𝐹)‘𝑦) = ((𝐾 ∘ 𝐹)‘𝑦)))
23		eqfnfv 5616	. . 3 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
24	15, 18, 23	syl2anc 411	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
25	14, 22, 24	3bitr4d 220	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ 𝐻 = 𝐾))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455   ∘ ccom 4632   Fn wfn 5213  ⟶wf 5214  –onto→wfo 5216  ‘cfv 5218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fo 5224  df-fv 5226

This theorem is referenced by:  mapen  6849  hashfacen  10819