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Theorem cocan1 7052
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 6761 . . . . . 6 ((𝐻:𝐴𝐵𝑥𝐴) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
213ad2antl2 1187 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
3 fvco3 6761 . . . . . 6 ((𝐾:𝐴𝐵𝑥𝐴) → ((𝐹𝐾)‘𝑥) = (𝐹‘(𝐾𝑥)))
433ad2antl3 1188 . . . . 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 1192 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → 𝐹:𝐵1-1𝐶)
7 ffvelrn 6853 . . . . . 6 ((𝐻:𝐴𝐵𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
873ad2antl2 1187 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
9 ffvelrn 6853 . . . . . 6 ((𝐾:𝐴𝐵𝑥𝐴) → (𝐾𝑥) ∈ 𝐵)
1093ad2antl3 1188 . . . . 5 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (𝐾𝑥) ∈ 𝐵)
11 f1fveq 7025 . . . . 5 ((𝐹:𝐵1-1𝐶 ∧ ((𝐻𝑥) ∈ 𝐵 ∧ (𝐾𝑥) ∈ 𝐵)) → ((𝐹‘(𝐻𝑥)) = (𝐹‘(𝐾𝑥)) ↔ (𝐻𝑥) = (𝐾𝑥)))
126, 8, 10, 11syl12anc 836 . . . 4 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → ((𝐹‘(𝐻𝑥)) = (𝐹‘(𝐾𝑥)) ↔ (𝐻𝑥) = (𝐾𝑥)))
135, 12bitrd 282 . . 3 (((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) ∧ 𝑥𝐴) → (((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥) ↔ (𝐻𝑥) = (𝐾𝑥)))
1413ralbidva 3108 . 2 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (∀𝑥𝐴 ((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐾𝑥)))
15 f1f 6568 . . . . . 6 (𝐹:𝐵1-1𝐶𝐹:𝐵𝐶)
16153ad2ant1 1134 . . . . 5 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐹:𝐵𝐶)
1716ffnd 6499 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐹 Fn 𝐵)
18 simp2 1138 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐻:𝐴𝐵)
19 fnfco 6537 . . . 4 ((𝐹 Fn 𝐵𝐻:𝐴𝐵) → (𝐹𝐻) Fn 𝐴)
2017, 18, 19syl2anc 587 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (𝐹𝐻) Fn 𝐴)
21 simp3 1139 . . . 4 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐾:𝐴𝐵)
22 fnfco 6537 . . . 4 ((𝐹 Fn 𝐵𝐾:𝐴𝐵) → (𝐹𝐾) Fn 𝐴)
2317, 21, 22syl2anc 587 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (𝐹𝐾) Fn 𝐴)
24 eqfnfv 6803 . . 3 (((𝐹𝐻) Fn 𝐴 ∧ (𝐹𝐾) Fn 𝐴) → ((𝐹𝐻) = (𝐹𝐾) ↔ ∀𝑥𝐴 ((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥)))
2520, 23, 24syl2anc 587 . 2 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → ((𝐹𝐻) = (𝐹𝐾) ↔ ∀𝑥𝐴 ((𝐹𝐻)‘𝑥) = ((𝐹𝐾)‘𝑥)))
2618ffnd 6499 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐻 Fn 𝐴)
2721ffnd 6499 . . 3 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → 𝐾 Fn 𝐴)
28 eqfnfv 6803 . . 3 ((𝐻 Fn 𝐴𝐾 Fn 𝐴) → (𝐻 = 𝐾 ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐾𝑥)))
2926, 27, 28syl2anc 587 . 2 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐾𝑥)))
3014, 25, 293bitr4d 314 1 ((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → ((𝐹𝐻) = (𝐹𝐾) ↔ 𝐻 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2113  wral 3053  ccom 5523   Fn wfn 6328  wf 6329  1-1wf1 6330  cfv 6333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fv 6341
This theorem is referenced by:  mapen  8724  mapfien  8938  hashfacen  13897  hashfacenOLD  13898  setcmon  17452  derangenlem  32696  subfacp1lem5  32709
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