MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cocan2 Structured version   Visualization version   GIF version

Theorem cocan2 7301
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:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → ((𝐻𝐹) = (𝐾𝐹) ↔ 𝐻 = 𝐾))

Proof of Theorem cocan2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fof 6810 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
213ad2ant1 1130 . . . . . 6 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐹:𝐴𝐵)
3 fvco3 6996 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
42, 3sylan 578 . . . . 5 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
5 fvco3 6996 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐾𝐹)‘𝑦) = (𝐾‘(𝐹𝑦)))
62, 5sylan 578 . . . . 5 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → ((𝐾𝐹)‘𝑦) = (𝐾‘(𝐹𝑦)))
74, 6eqeq12d 2741 . . . 4 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → (((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦))))
87ralbidva 3165 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ ∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦))))
9 fveq2 6896 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐻‘(𝐹𝑦)) = (𝐻𝑥))
10 fveq2 6896 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐾‘(𝐹𝑦)) = (𝐾𝑥))
119, 10eqeq12d 2741 . . . . 5 ((𝐹𝑦) = 𝑥 → ((𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ (𝐻𝑥) = (𝐾𝑥)))
1211cbvfo 7298 . . . 4 (𝐹:𝐴onto𝐵 → (∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
13123ad2ant1 1130 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
148, 13bitrd 278 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
15 simp2 1134 . . . 4 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐻 Fn 𝐵)
16 fnfco 6762 . . . 4 ((𝐻 Fn 𝐵𝐹:𝐴𝐵) → (𝐻𝐹) Fn 𝐴)
1715, 2, 16syl2anc 582 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻𝐹) Fn 𝐴)
18 simp3 1135 . . . 4 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
19 fnfco 6762 . . . 4 ((𝐾 Fn 𝐵𝐹:𝐴𝐵) → (𝐾𝐹) Fn 𝐴)
2018, 2, 19syl2anc 582 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐾𝐹) Fn 𝐴)
21 eqfnfv 7039 . . 3 (((𝐻𝐹) Fn 𝐴 ∧ (𝐾𝐹) Fn 𝐴) → ((𝐻𝐹) = (𝐾𝐹) ↔ ∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦)))
2217, 20, 21syl2anc 582 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → ((𝐻𝐹) = (𝐾𝐹) ↔ ∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦)))
23 eqfnfv 7039 . . 3 ((𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
2415, 18, 23syl2anc 582 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
2514, 22, 243bitr4d 310 1 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → ((𝐻𝐹) = (𝐾𝐹) ↔ 𝐻 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  ccom 5682   Fn wfn 6544  wf 6545  ontowfo 6547  cfv 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fo 6555  df-fv 6557
This theorem is referenced by:  mapen  9169  mapfien  9438  hashfacen  14457  hashfacenOLD  14458  setcepi  18096  qtopeu  23681  qtophmeo  23782  fmptco1f1o  32519  1arithidomlem2  33369  derangenlem  34932
  Copyright terms: Public domain W3C validator