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Theorem cocanfo 32682
 Description: Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
cocanfo (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐺 = 𝐻)

Proof of Theorem cocanfo
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 788 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → (𝐺𝐹) = (𝐻𝐹))
21fveq1d 6105 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → ((𝐺𝐹)‘𝑦) = ((𝐻𝐹)‘𝑦))
3 simpl1 1057 . . . . . . 7 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐹:𝐴onto𝐵)
4 fof 6028 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
53, 4syl 17 . . . . . 6 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐹:𝐴𝐵)
6 fvco3 6185 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦)))
75, 6sylan 487 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦)))
8 fvco3 6185 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
95, 8sylan 487 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
102, 7, 93eqtr3d 2652 . . . 4 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)))
1110ralrimiva 2949 . . 3 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → ∀𝑦𝐴 (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)))
12 fveq2 6103 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐺‘(𝐹𝑦)) = (𝐺𝑥))
13 fveq2 6103 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐻‘(𝐹𝑦)) = (𝐻𝑥))
1412, 13eqeq12d 2625 . . . . 5 ((𝐹𝑦) = 𝑥 → ((𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)) ↔ (𝐺𝑥) = (𝐻𝑥)))
1514cbvfo 6444 . . . 4 (𝐹:𝐴onto𝐵 → (∀𝑦𝐴 (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
163, 15syl 17 . . 3 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → (∀𝑦𝐴 (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
1711, 16mpbid 221 . 2 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥))
18 eqfnfv 6219 . . . 4 ((𝐺 Fn 𝐵𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
19183adant1 1072 . . 3 ((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
2019adantr 480 . 2 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → (𝐺 = 𝐻 ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
2117, 20mpbird 246 1 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐺 = 𝐻)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812 This theorem is referenced by: (None)
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