Theorem cocan2 7026

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 6565	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	3ad2ant1 1130	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐹:𝐴⟶𝐵)
3		fvco3 6737	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
4	2, 3	sylan 583	. . . . 5 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
5		fvco3 6737	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑦) = (𝐾‘(𝐹‘𝑦)))
6	2, 5	sylan 583	. . . . 5 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑦) = (𝐾‘(𝐹‘𝑦)))
7	4, 6	eqeq12d 2814	. . . 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 3161	. . 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 6645	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐻‘(𝐹‘𝑦)) = (𝐻‘𝑥))
10		fveq2 6645	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐾‘(𝐹‘𝑦)) = (𝐾‘𝑥))
11	9, 10	eqeq12d 2814	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 → ((𝐻‘(𝐹‘𝑦)) = (𝐾‘(𝐹‘𝑦)) ↔ (𝐻‘𝑥) = (𝐾‘𝑥)))
12	11	cbvfo 7023	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ 𝐴 (𝐻‘(𝐹‘𝑦)) = (𝐾‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
13	12	3ad2ant1 1130	. . 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 282	. 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 1134	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐻 Fn 𝐵)
16		fnfco 6517	. . . 4 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐻 ∘ 𝐹) Fn 𝐴)
17	15, 2, 16	syl2anc 587	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 ∘ 𝐹) Fn 𝐴)
18		simp3 1135	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
19		fnfco 6517	. . . 4 ⊢ ((𝐾 Fn 𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐾 ∘ 𝐹) Fn 𝐴)
20	18, 2, 19	syl2anc 587	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐾 ∘ 𝐹) Fn 𝐴)
21		eqfnfv 6779	. . 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 587	. 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 6779	. . 3 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
24	15, 18, 23	syl2anc 587	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥 ∈ 𝐵 (𝐻‘𝑥) = (𝐾‘𝑥)))
25	14, 22, 24	3bitr4d 314	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ 𝐻 = 𝐾))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∘ ccom 5523   Fn wfn 6319  ⟶wf 6320  –onto→wfo 6322  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332

This theorem is referenced by:  mapen  8665  mapfien  8855  hashfacen  13808  setcepi  17340  qtopeu  22321  qtophmeo  22422  fmptco1f1o  30392  derangenlem  32531