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Theorem cocan1 7311
Description: An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
cocan1 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → ((𝐹𝐻) = (𝐹𝐾) ↔ 𝐻 = 𝐾))

Proof of Theorem cocan1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fvco3 7008 . . . . . 6 ((𝐻:𝐴𝐵𝑥𝐴) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
213ad2antl2 1185 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
3 fvco3 7008 . . . . . 6 ((𝐾:𝐴𝐵𝑥𝐴) → ((𝐹𝐾)‘𝑥) = (𝐹‘(𝐾𝑥)))
433ad2antl3 1186 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → ((𝐹𝐾)‘𝑥) = (𝐹‘(𝐾𝑥)))
52, 4eqeq12d 2751 . . . 4 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥) ↔ (𝐹‘(𝐻𝑥)) = (𝐹‘(𝐾𝑥))))
6 simpl1 1190 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → 𝐹:𝐵1-1𝐶)
7 ffvelcdm 7101 . . . . . 6 ((𝐻:𝐴𝐵𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
873ad2antl2 1185 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
9 ffvelcdm 7101 . . . . . 6 ((𝐾:𝐴𝐵𝑥𝐴) → (𝐾𝑥) ∈ 𝐵)
1093ad2antl3 1186 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (𝐾𝑥) ∈ 𝐵)
11 f1fveq 7282 . . . . 5 ((𝐹:𝐵1-1𝐶 ∧ ((𝐻𝑥) ∈ 𝐵 ∧ (𝐾𝑥) ∈ 𝐵)) → ((𝐹‘(𝐻𝑥)) = (𝐹‘(𝐾𝑥)) ↔ (𝐻𝑥) = (𝐾𝑥)))
126, 8, 10, 11syl12anc 837 . . . 4 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → ((𝐹‘(𝐻𝑥)) = (𝐹‘(𝐾𝑥)) ↔ (𝐻𝑥) = (𝐾𝑥)))
135, 12bitrd 279 . . 3 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥) ↔ (𝐻𝑥) = (𝐾𝑥)))
1413ralbidva 3174 . 2 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (∀𝑥𝐴 ((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐾𝑥)))
15 f1f 6805 . . . . . 6 (𝐹:𝐵1-1𝐶𝐹:𝐵𝐶)
16153ad2ant1 1132 . . . . 5 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐹:𝐵𝐶)
1716ffnd 6738 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐹 Fn 𝐵)
18 simp2 1136 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐻:𝐴𝐵)
19 fnfco 6774 . . . 4 ((𝐹 Fn 𝐵𝐻:𝐴𝐵) → (𝐹𝐻) Fn 𝐴)
2017, 18, 19syl2anc 584 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (𝐹𝐻) Fn 𝐴)
21 simp3 1137 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐾:𝐴𝐵)
22 fnfco 6774 . . . 4 ((𝐹 Fn 𝐵𝐾:𝐴𝐵) → (𝐹𝐾) Fn 𝐴)
2317, 21, 22syl2anc 584 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (𝐹𝐾) Fn 𝐴)
24 eqfnfv 7051 . . 3 (((𝐹𝐻) Fn 𝐴 ∧ (𝐹𝐾) Fn 𝐴) → ((𝐹𝐻) = (𝐹𝐾) ↔ ∀𝑥𝐴 ((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥)))
2520, 23, 24syl2anc 584 . 2 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → ((𝐹𝐻) = (𝐹𝐾) ↔ ∀𝑥𝐴 ((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥)))
2618ffnd 6738 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐻 Fn 𝐴)
2721ffnd 6738 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐾 Fn 𝐴)
28 eqfnfv 7051 . . 3 ((𝐻 Fn 𝐴𝐾 Fn 𝐴) → (𝐻 = 𝐾 ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐾𝑥)))
2926, 27, 28syl2anc 584 . 2 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐾𝑥)))
3014, 25, 293bitr4d 311 1 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → ((𝐹𝐻) = (𝐹𝐾) ↔ 𝐻 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  ccom 5693   Fn wfn 6558  wf 6559  1-1wf1 6560  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571
This theorem is referenced by:  mapen  9180  mapfien  9446  hashfacen  14490  setcmon  18141  derangenlem  35156  subfacp1lem5  35169
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