Theorem cocan2 5984

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 5610	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	3ad2ant1 1049	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐹:𝐴⟶𝐵)
3		fvco3 5770	. . . . . 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 5770	. . . . . 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 2253	. . . 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 2546	. . 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 5690	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐻‘(𝐹‘𝑦)) = (𝐻‘𝑥))
10		fveq2 5690	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐾‘(𝐹‘𝑦)) = (𝐾‘𝑥))
11	9, 10	eqeq12d 2253	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 → ((𝐻‘(𝐹‘𝑦)) = (𝐾‘(𝐹‘𝑦)) ↔ (𝐻‘𝑥) = (𝐾‘𝑥)))
12	11	cbvfo 5981	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ 𝐴 (𝐻‘(𝐹‘𝑦)) = (𝐾‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
13	12	3ad2ant1 1049	. . 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 1029	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐻 Fn 𝐵)
16		fnfco 5559	. . . 4 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐻 ∘ 𝐹) Fn 𝐴)
17	15, 2, 16	syl2anc 415	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 ∘ 𝐹) Fn 𝐴)
18		simp3 1030	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
19		fnfco 5559	. . . 4 ⊢ ((𝐾 Fn 𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐾 ∘ 𝐹) Fn 𝐴)
20	18, 2, 19	syl2anc 415	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐾 ∘ 𝐹) Fn 𝐴)
21		eqfnfv 5797	. . 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 415	. 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 5797	. . 3 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
24	15, 18, 23	syl2anc 415	. 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 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ∘ ccom 4773   Fn wfn 5367  ⟶wf 5368  –onto→wfo 5370  ‘cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380

This theorem is referenced by:  mapen  7136  hashfacen  11262