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Theorem cocan2 6693
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 6257 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
213ad2ant1 1127 . . . . . 6 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐹:𝐴𝐵)
3 fvco3 6419 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
42, 3sylan 569 . . . . 5 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
5 fvco3 6419 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐾𝐹)‘𝑦) = (𝐾‘(𝐹𝑦)))
62, 5sylan 569 . . . . 5 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → ((𝐾𝐹)‘𝑦) = (𝐾‘(𝐹𝑦)))
74, 6eqeq12d 2786 . . . 4 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → (((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦))))
87ralbidva 3134 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ ∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦))))
9 fveq2 6333 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐻‘(𝐹𝑦)) = (𝐻𝑥))
10 fveq2 6333 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐾‘(𝐹𝑦)) = (𝐾𝑥))
119, 10eqeq12d 2786 . . . . 5 ((𝐹𝑦) = 𝑥 → ((𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ (𝐻𝑥) = (𝐾𝑥)))
1211cbvfo 6690 . . . 4 (𝐹:𝐴onto𝐵 → (∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
13123ad2ant1 1127 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
148, 13bitrd 268 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
15 simp2 1131 . . . 4 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐻 Fn 𝐵)
16 fnfco 6210 . . . 4 ((𝐻 Fn 𝐵𝐹:𝐴𝐵) → (𝐻𝐹) Fn 𝐴)
1715, 2, 16syl2anc 573 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻𝐹) Fn 𝐴)
18 simp3 1132 . . . 4 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
19 fnfco 6210 . . . 4 ((𝐾 Fn 𝐵𝐹:𝐴𝐵) → (𝐾𝐹) Fn 𝐴)
2018, 2, 19syl2anc 573 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐾𝐹) Fn 𝐴)
21 eqfnfv 6456 . . 3 (((𝐻𝐹) Fn 𝐴 ∧ (𝐾𝐹) Fn 𝐴) → ((𝐻𝐹) = (𝐾𝐹) ↔ ∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦)))
2217, 20, 21syl2anc 573 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → ((𝐻𝐹) = (𝐾𝐹) ↔ ∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦)))
23 eqfnfv 6456 . . 3 ((𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
2415, 18, 23syl2anc 573 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
2514, 22, 243bitr4d 300 1 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → ((𝐻𝐹) = (𝐾𝐹) ↔ 𝐻 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  ccom 5254   Fn wfn 6025  wf 6026  ontowfo 6028  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fo 6036  df-fv 6038
This theorem is referenced by:  mapen  8284  mapfien  8473  hashfacen  13440  setcepi  16945  qtopeu  21740  qtophmeo  21841  fmptco1f1o  29774  derangenlem  31491
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