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Theorem cocan1 6586
 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 6314 . . . . . 6 ((𝐻:𝐴𝐵𝑥𝐴) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
213ad2antl2 1244 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
3 fvco3 6314 . . . . . 6 ((𝐾:𝐴𝐵𝑥𝐴) → ((𝐹𝐾)‘𝑥) = (𝐹‘(𝐾𝑥)))
433ad2antl3 1245 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → ((𝐹𝐾)‘𝑥) = (𝐹‘(𝐾𝑥)))
52, 4eqeq12d 2666 . . . 4 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥) ↔ (𝐹‘(𝐻𝑥)) = (𝐹‘(𝐾𝑥))))
6 simpl1 1084 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → 𝐹:𝐵1-1𝐶)
7 ffvelrn 6397 . . . . . 6 ((𝐻:𝐴𝐵𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
873ad2antl2 1244 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
9 ffvelrn 6397 . . . . . 6 ((𝐾:𝐴𝐵𝑥𝐴) → (𝐾𝑥) ∈ 𝐵)
1093ad2antl3 1245 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (𝐾𝑥) ∈ 𝐵)
11 f1fveq 6559 . . . . 5 ((𝐹:𝐵1-1𝐶 ∧ ((𝐻𝑥) ∈ 𝐵 ∧ (𝐾𝑥) ∈ 𝐵)) → ((𝐹‘(𝐻𝑥)) = (𝐹‘(𝐾𝑥)) ↔ (𝐻𝑥) = (𝐾𝑥)))
126, 8, 10, 11syl12anc 1364 . . . 4 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → ((𝐹‘(𝐻𝑥)) = (𝐹‘(𝐾𝑥)) ↔ (𝐻𝑥) = (𝐾𝑥)))
135, 12bitrd 268 . . 3 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥) ↔ (𝐻𝑥) = (𝐾𝑥)))
1413ralbidva 3014 . 2 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (∀𝑥𝐴 ((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐾𝑥)))
15 f1f 6139 . . . . . 6 (𝐹:𝐵1-1𝐶𝐹:𝐵𝐶)
16153ad2ant1 1102 . . . . 5 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐹:𝐵𝐶)
17 ffn 6083 . . . . 5 (𝐹:𝐵𝐶𝐹 Fn 𝐵)
1816, 17syl 17 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐹 Fn 𝐵)
19 simp2 1082 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐻:𝐴𝐵)
20 fnfco 6107 . . . 4 ((𝐹 Fn 𝐵𝐻:𝐴𝐵) → (𝐹𝐻) Fn 𝐴)
2118, 19, 20syl2anc 694 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (𝐹𝐻) Fn 𝐴)
22 simp3 1083 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐾:𝐴𝐵)
23 fnfco 6107 . . . 4 ((𝐹 Fn 𝐵𝐾:𝐴𝐵) → (𝐹𝐾) Fn 𝐴)
2418, 22, 23syl2anc 694 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (𝐹𝐾) Fn 𝐴)
25 eqfnfv 6351 . . 3 (((𝐹𝐻) Fn 𝐴 ∧ (𝐹𝐾) Fn 𝐴) → ((𝐹𝐻) = (𝐹𝐾) ↔ ∀𝑥𝐴 ((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥)))
2621, 24, 25syl2anc 694 . 2 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → ((𝐹𝐻) = (𝐹𝐾) ↔ ∀𝑥𝐴 ((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥)))
27 ffn 6083 . . . 4 (𝐻:𝐴𝐵𝐻 Fn 𝐴)
2819, 27syl 17 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐻 Fn 𝐴)
29 ffn 6083 . . . 4 (𝐾:𝐴𝐵𝐾 Fn 𝐴)
3022, 29syl 17 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐾 Fn 𝐴)
31 eqfnfv 6351 . . 3 ((𝐻 Fn 𝐴𝐾 Fn 𝐴) → (𝐻 = 𝐾 ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐾𝑥)))
3228, 30, 31syl2anc 694 . 2 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐾𝑥)))
3314, 26, 323bitr4d 300 1 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → ((𝐹𝐻) = (𝐹𝐾) ↔ 𝐻 = 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941   ∘ ccom 5147   Fn wfn 5921  ⟶wf 5922  –1-1→wf1 5923  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fv 5934 This theorem is referenced by:  mapen  8165  mapfien  8354  hashfacen  13276  setcmon  16784  derangenlem  31279  subfacp1lem5  31292
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