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Theorem cocan1 7163
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 6867 . . . . . 6 ((𝐻:𝐴𝐵𝑥𝐴) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
213ad2antl2 1185 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
3 fvco3 6867 . . . . . 6 ((𝐾:𝐴𝐵𝑥𝐴) → ((𝐹𝐾)‘𝑥) = (𝐹‘(𝐾𝑥)))
433ad2antl3 1186 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → ((𝐹𝐾)‘𝑥) = (𝐹‘(𝐾𝑥)))
52, 4eqeq12d 2754 . . . 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 ffvelrn 6959 . . . . . 6 ((𝐻:𝐴𝐵𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
873ad2antl2 1185 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
9 ffvelrn 6959 . . . . . 6 ((𝐾:𝐴𝐵𝑥𝐴) → (𝐾𝑥) ∈ 𝐵)
1093ad2antl3 1186 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (𝐾𝑥) ∈ 𝐵)
11 f1fveq 7135 . . . . 5 ((𝐹:𝐵1-1𝐶 ∧ ((𝐻𝑥) ∈ 𝐵 ∧ (𝐾𝑥) ∈ 𝐵)) → ((𝐹‘(𝐻𝑥)) = (𝐹‘(𝐾𝑥)) ↔ (𝐻𝑥) = (𝐾𝑥)))
126, 8, 10, 11syl12anc 834 . . . 4 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → ((𝐹‘(𝐻𝑥)) = (𝐹‘(𝐾𝑥)) ↔ (𝐻𝑥) = (𝐾𝑥)))
135, 12bitrd 278 . . 3 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥) ↔ (𝐻𝑥) = (𝐾𝑥)))
1413ralbidva 3111 . 2 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (∀𝑥𝐴 ((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐾𝑥)))
15 f1f 6670 . . . . . 6 (𝐹:𝐵1-1𝐶𝐹:𝐵𝐶)
16153ad2ant1 1132 . . . . 5 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐹:𝐵𝐶)
1716ffnd 6601 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐹 Fn 𝐵)
18 simp2 1136 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐻:𝐴𝐵)
19 fnfco 6639 . . . 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 6639 . . . 4 ((𝐹 Fn 𝐵𝐾:𝐴𝐵) → (𝐹𝐾) Fn 𝐴)
2317, 21, 22syl2anc 584 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (𝐹𝐾) Fn 𝐴)
24 eqfnfv 6909 . . 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 6601 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐻 Fn 𝐴)
2721ffnd 6601 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐾 Fn 𝐴)
28 eqfnfv 6909 . . 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 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  ccom 5593   Fn wfn 6428  wf 6429  1-1wf1 6430  cfv 6433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fv 6441
This theorem is referenced by:  mapen  8928  mapfien  9167  hashfacen  14166  hashfacenOLD  14167  setcmon  17802  derangenlem  33133  subfacp1lem5  33146
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