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Theorem cocan2 5455
Description: A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
cocan2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → ((𝐻𝐹) = (𝐾𝐹) ↔ 𝐻 = 𝐾))

Proof of Theorem cocan2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fof 5133 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
213ad2ant1 936 . . . . . 6 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐹:𝐴𝐵)
3 fvco3 5271 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
42, 3sylan 271 . . . . 5 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
5 fvco3 5271 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐾𝐹)‘𝑦) = (𝐾‘(𝐹𝑦)))
62, 5sylan 271 . . . . 5 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → ((𝐾𝐹)‘𝑦) = (𝐾‘(𝐹𝑦)))
74, 6eqeq12d 2070 . . . 4 (((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) ∧ 𝑦𝐴) → (((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦))))
87ralbidva 2339 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ ∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦))))
9 fveq2 5205 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐻‘(𝐹𝑦)) = (𝐻𝑥))
10 fveq2 5205 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐾‘(𝐹𝑦)) = (𝐾𝑥))
119, 10eqeq12d 2070 . . . . 5 ((𝐹𝑦) = 𝑥 → ((𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ (𝐻𝑥) = (𝐾𝑥)))
1211cbvfo 5452 . . . 4 (𝐹:𝐴onto𝐵 → (∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
13123ad2ant1 936 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 (𝐻‘(𝐹𝑦)) = (𝐾‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
148, 13bitrd 181 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦) ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
15 simp2 916 . . . 4 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐻 Fn 𝐵)
16 fnfco 5092 . . . 4 ((𝐻 Fn 𝐵𝐹:𝐴𝐵) → (𝐻𝐹) Fn 𝐴)
1715, 2, 16syl2anc 397 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻𝐹) Fn 𝐴)
18 simp3 917 . . . 4 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
19 fnfco 5092 . . . 4 ((𝐾 Fn 𝐵𝐹:𝐴𝐵) → (𝐾𝐹) Fn 𝐴)
2018, 2, 19syl2anc 397 . . 3 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐾𝐹) Fn 𝐴)
21 eqfnfv 5292 . . 3 (((𝐻𝐹) Fn 𝐴 ∧ (𝐾𝐹) Fn 𝐴) → ((𝐻𝐹) = (𝐾𝐹) ↔ ∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦)))
2217, 20, 21syl2anc 397 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → ((𝐻𝐹) = (𝐾𝐹) ↔ ∀𝑦𝐴 ((𝐻𝐹)‘𝑦) = ((𝐾𝐹)‘𝑦)))
23 eqfnfv 5292 . . 3 ((𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
2415, 18, 23syl2anc 397 . 2 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → (𝐻 = 𝐾 ↔ ∀𝑥𝐵 (𝐻𝑥) = (𝐾𝑥)))
2514, 22, 243bitr4d 213 1 ((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → ((𝐻𝐹) = (𝐾𝐹) ↔ 𝐻 = 𝐾))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  wral 2323  ccom 4376   Fn wfn 4924  wf 4925  ontowfo 4927  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fo 4935  df-fv 4937
This theorem is referenced by: (None)
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