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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 6811 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
213ad2ant1 1131 . . . . . 6 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐹:𝐴𝐵)
3 fvco3 6997 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
42, 3sylan 579 . . . . 5 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
5 fvco3 6997 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐾𝐹)‘𝑦) = (𝐾‘(𝐹𝑦)))
62, 5sylan 579 . . . . 5 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → ((𝐾𝐹)‘𝑦) = (𝐾‘(𝐹𝑦)))
74, 6eqeq12d 2744 . . . 4 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → (((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦))))
87ralbidva 3172 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ ∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦))))
9 fveq2 6897 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐻‘(𝐹𝑦)) = (𝐻𝑥))
10 fveq2 6897 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐾‘(𝐹𝑦)) = (𝐾𝑥))
119, 10eqeq12d 2744 . . . . 5 ((𝐹𝑦) = 𝑥 → ((𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ (𝐻𝑥) = (𝐾𝑥)))
1211cbvfo 7298 . . . 4 (𝐹:𝐴onto𝐵 → (∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
13123ad2ant1 1131 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
148, 13bitrd 279 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
15 simp2 1135 . . . 4 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐻 Fn 𝐵)
16 fnfco 6762 . . . 4 ((𝐻 Fn 𝐵𝐹:𝐴𝐵) → (𝐻𝐹) Fn 𝐴)
1715, 2, 16syl2anc 583 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻𝐹) Fn 𝐴)
18 simp3 1136 . . . 4 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
19 fnfco 6762 . . . 4 ((𝐾 Fn 𝐵𝐹:𝐴𝐵) → (𝐾𝐹) Fn 𝐴)
2018, 2, 19syl2anc 583 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐾𝐹) Fn 𝐴)
21 eqfnfv 7040 . . 3 (((𝐻𝐹) Fn 𝐴 ∧ (𝐾𝐹) Fn 𝐴) → ((𝐻𝐹) = (𝐾𝐹) ↔ ∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦)))
2217, 20, 21syl2anc 583 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → ((𝐻𝐹) = (𝐾𝐹) ↔ ∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦)))
23 eqfnfv 7040 . . 3 ((𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
2415, 18, 23syl2anc 583 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
2514, 22, 243bitr4d 311 1 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → ((𝐻𝐹) = (𝐾𝐹) ↔ 𝐻 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3058  ccom 5682   Fn wfn 6543  wf 6544  ontowfo 6546  cfv 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  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 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fo 6554  df-fv 6556
This theorem is referenced by:  mapen  9166  mapfien  9432  hashfacen  14446  hashfacenOLD  14447  setcepi  18077  qtopeu  23633  qtophmeo  23734  fmptco1f1o  32431  derangenlem  34781
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