Theorem cocan2 5767

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 5420	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	3ad2ant1 1013	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐹:𝐴⟶𝐵)
3		fvco3 5567	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
4	2, 3	sylan 281	. . . . 5 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
5		fvco3 5567	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑦) = (𝐾‘(𝐹‘𝑦)))
6	2, 5	sylan 281	. . . . 5 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑦) = (𝐾‘(𝐹‘𝑦)))
7	4, 6	eqeq12d 2185	. . . 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 2466	. . 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 5496	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐻‘(𝐹‘𝑦)) = (𝐻‘𝑥))
10		fveq2 5496	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐾‘(𝐹‘𝑦)) = (𝐾‘𝑥))
11	9, 10	eqeq12d 2185	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 → ((𝐻‘(𝐹‘𝑦)) = (𝐾‘(𝐹‘𝑦)) ↔ (𝐻‘𝑥) = (𝐾‘𝑥)))
12	11	cbvfo 5764	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ 𝐴 (𝐻‘(𝐹‘𝑦)) = (𝐾‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
13	12	3ad2ant1 1013	. . 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 187	. 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 993	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐻 Fn 𝐵)
16		fnfco 5372	. . . 4 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐻 ∘ 𝐹) Fn 𝐴)
17	15, 2, 16	syl2anc 409	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 ∘ 𝐹) Fn 𝐴)
18		simp3 994	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
19		fnfco 5372	. . . 4 ⊢ ((𝐾 Fn 𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐾 ∘ 𝐹) Fn 𝐴)
20	18, 2, 19	syl2anc 409	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐾 ∘ 𝐹) Fn 𝐴)
21		eqfnfv 5593	. . 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 409	. 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 5593	. . 3 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
24	15, 18, 23	syl2anc 409	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
25	14, 22, 24	3bitr4d 219	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ 𝐻 = 𝐾))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  ∀wral 2448   ∘ ccom 4615   Fn wfn 5193  ⟶wf 5194  –onto→wfo 5196  ‘cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206

This theorem is referenced by:  mapen  6824  hashfacen  10771