ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cocan2 GIF version

Theorem cocan2 5657
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 5315 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
213ad2ant1 987 . . . . . 6 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐹:𝐴𝐵)
3 fvco3 5460 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
42, 3sylan 281 . . . . 5 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
5 fvco3 5460 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐾𝐹)‘𝑦) = (𝐾‘(𝐹𝑦)))
62, 5sylan 281 . . . . 5 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → ((𝐾𝐹)‘𝑦) = (𝐾‘(𝐹𝑦)))
74, 6eqeq12d 2132 . . . 4 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → (((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦))))
87ralbidva 2410 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ ∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦))))
9 fveq2 5389 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐻‘(𝐹𝑦)) = (𝐻𝑥))
10 fveq2 5389 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐾‘(𝐹𝑦)) = (𝐾𝑥))
119, 10eqeq12d 2132 . . . . 5 ((𝐹𝑦) = 𝑥 → ((𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ (𝐻𝑥) = (𝐾𝑥)))
1211cbvfo 5654 . . . 4 (𝐹:𝐴onto𝐵 → (∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
13123ad2ant1 987 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
148, 13bitrd 187 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
15 simp2 967 . . . 4 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐻 Fn 𝐵)
16 fnfco 5267 . . . 4 ((𝐻 Fn 𝐵𝐹:𝐴𝐵) → (𝐻𝐹) Fn 𝐴)
1715, 2, 16syl2anc 408 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻𝐹) Fn 𝐴)
18 simp3 968 . . . 4 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
19 fnfco 5267 . . . 4 ((𝐾 Fn 𝐵𝐹:𝐴𝐵) → (𝐾𝐹) Fn 𝐴)
2018, 2, 19syl2anc 408 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐾𝐹) Fn 𝐴)
21 eqfnfv 5486 . . 3 (((𝐻𝐹) Fn 𝐴 ∧ (𝐾𝐹) Fn 𝐴) → ((𝐻𝐹) = (𝐾𝐹) ↔ ∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦)))
2217, 20, 21syl2anc 408 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → ((𝐻𝐹) = (𝐾𝐹) ↔ ∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦)))
23 eqfnfv 5486 . . 3 ((𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
2415, 18, 23syl2anc 408 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
2514, 22, 243bitr4d 219 1 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → ((𝐻𝐹) = (𝐾𝐹) ↔ 𝐻 = 𝐾))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 947   = wceq 1316  wcel 1465  wral 2393  ccom 4513   Fn wfn 5088  wf 5089  ontowfo 5091  cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101
This theorem is referenced by:  mapen  6708  hashfacen  10547
  Copyright terms: Public domain W3C validator