Theorem cocan2 7236

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 6739	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	3ad2ant1 1139	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐹:𝐴⟶𝐵)
3		fvco3 6927	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
4	2, 3	sylan 586	. . . . 5 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
5		fvco3 6927	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑦) = (𝐾‘(𝐹‘𝑦)))
6	2, 5	sylan 586	. . . . 5 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑦) = (𝐾‘(𝐹‘𝑦)))
7	4, 6	eqeq12d 2755	. . . 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 3160	. . 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 6827	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐻‘(𝐹‘𝑦)) = (𝐻‘𝑥))
10		fveq2 6827	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐾‘(𝐹‘𝑦)) = (𝐾‘𝑥))
11	9, 10	eqeq12d 2755	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 → ((𝐻‘(𝐹‘𝑦)) = (𝐾‘(𝐹‘𝑦)) ↔ (𝐻‘𝑥) = (𝐾‘𝑥)))
12	11	cbvfo 7233	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ 𝐴 (𝐻‘(𝐹‘𝑦)) = (𝐾‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
13	12	3ad2ant1 1139	. . 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 280	. 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 1143	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐻 Fn 𝐵)
16		fnfco 6692	. . . 4 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐻 ∘ 𝐹) Fn 𝐴)
17	15, 2, 16	syl2anc 590	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 ∘ 𝐹) Fn 𝐴)
18		simp3 1144	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
19		fnfco 6692	. . . 4 ⊢ ((𝐾 Fn 𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐾 ∘ 𝐹) Fn 𝐴)
20	18, 2, 19	syl2anc 590	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐾 ∘ 𝐹) Fn 𝐴)
21		eqfnfv 6971	. . 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 590	. 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 6971	. . 3 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
24	15, 18, 23	syl2anc 590	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
25	14, 22, 24	3bitr4d 312	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ 𝐻 = 𝐾))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ∘ ccom 5622   Fn wfn 6480  ⟶wf 6481  –onto→wfo 6483  ‘cfv 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fo 6491  df-fv 6493

This theorem is referenced by:  mapen  9069  mapfien  9311  hashfacen  14407  setcepi  18046  qtopeu  23699  qtophmeo  23800  fmptco1f1o  32725  1arithidomlem2  33619  derangenlem  35399