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Theorem cocan1 6701
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 6429 . . . . . 6 ((𝐻:𝐴𝐵𝑥𝐴) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
213ad2antl2 1199 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
3 fvco3 6429 . . . . . 6 ((𝐾:𝐴𝐵𝑥𝐴) → ((𝐹𝐾)‘𝑥) = (𝐹‘(𝐾𝑥)))
433ad2antl3 1200 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → ((𝐹𝐾)‘𝑥) = (𝐹‘(𝐾𝑥)))
52, 4eqeq12d 2767 . . . 4 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥) ↔ (𝐹‘(𝐻𝑥)) = (𝐹‘(𝐾𝑥))))
6 simpl1 1225 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → 𝐹:𝐵1-1𝐶)
7 ffvelrn 6512 . . . . . 6 ((𝐻:𝐴𝐵𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
873ad2antl2 1199 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
9 ffvelrn 6512 . . . . . 6 ((𝐾:𝐴𝐵𝑥𝐴) → (𝐾𝑥) ∈ 𝐵)
1093ad2antl3 1200 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (𝐾𝑥) ∈ 𝐵)
11 f1fveq 6674 . . . . 5 ((𝐹:𝐵1-1𝐶 ∧ ((𝐻𝑥) ∈ 𝐵 ∧ (𝐾𝑥) ∈ 𝐵)) → ((𝐹‘(𝐻𝑥)) = (𝐹‘(𝐾𝑥)) ↔ (𝐻𝑥) = (𝐾𝑥)))
126, 8, 10, 11syl12anc 1471 . . . 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 3115 . 2 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (∀𝑥𝐴 ((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐾𝑥)))
15 f1f 6254 . . . . . 6 (𝐹:𝐵1-1𝐶𝐹:𝐵𝐶)
16153ad2ant1 1127 . . . . 5 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐹:𝐵𝐶)
17 ffn 6198 . . . . 5 (𝐹:𝐵𝐶𝐹 Fn 𝐵)
1816, 17syl 17 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐹 Fn 𝐵)
19 simp2 1131 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐻:𝐴𝐵)
20 fnfco 6222 . . . 4 ((𝐹 Fn 𝐵𝐻:𝐴𝐵) → (𝐹𝐻) Fn 𝐴)
2118, 19, 20syl2anc 696 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (𝐹𝐻) Fn 𝐴)
22 simp3 1132 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐾:𝐴𝐵)
23 fnfco 6222 . . . 4 ((𝐹 Fn 𝐵𝐾:𝐴𝐵) → (𝐹𝐾) Fn 𝐴)
2418, 22, 23syl2anc 696 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (𝐹𝐾) Fn 𝐴)
25 eqfnfv 6466 . . 3 (((𝐹𝐻) Fn 𝐴 ∧ (𝐹𝐾) Fn 𝐴) → ((𝐹𝐻) = (𝐹𝐾) ↔ ∀𝑥𝐴 ((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥)))
2621, 24, 25syl2anc 696 . 2 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → ((𝐹𝐻) = (𝐹𝐾) ↔ ∀𝑥𝐴 ((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥)))
27 ffn 6198 . . . 4 (𝐻:𝐴𝐵𝐻 Fn 𝐴)
2819, 27syl 17 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐻 Fn 𝐴)
29 ffn 6198 . . . 4 (𝐾:𝐴𝐵𝐾 Fn 𝐴)
3022, 29syl 17 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐾 Fn 𝐴)
31 eqfnfv 6466 . . 3 ((𝐻 Fn 𝐴𝐾 Fn 𝐴) → (𝐻 = 𝐾 ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐾𝑥)))
3228, 30, 31syl2anc 696 . 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 1072   = wceq 1624  wcel 2131  wral 3042  ccom 5262   Fn wfn 6036  wf 6037  1-1wf1 6038  cfv 6041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fv 6049
This theorem is referenced by:  mapen  8281  mapfien  8470  hashfacen  13422  setcmon  16930  derangenlem  31452  subfacp1lem5  31465
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