Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  f1ocan1fv Structured version   Visualization version   GIF version

Theorem f1ocan1fv 32691
 Description: Cancel a composition by a bijection by preapplying the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
f1ocan1fv ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))

Proof of Theorem f1ocan1fv
StepHypRef Expression
1 f1of 6050 . . . 4 (𝐺:𝐴1-1-onto𝐵𝐺:𝐴𝐵)
213ad2ant2 1076 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → 𝐺:𝐴𝐵)
3 f1ocnv 6062 . . . . . 6 (𝐺:𝐴1-1-onto𝐵𝐺:𝐵1-1-onto𝐴)
4 f1of 6050 . . . . . 6 (𝐺:𝐵1-1-onto𝐴𝐺:𝐵𝐴)
53, 4syl 17 . . . . 5 (𝐺:𝐴1-1-onto𝐵𝐺:𝐵𝐴)
653ad2ant2 1076 . . . 4 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → 𝐺:𝐵𝐴)
7 simp3 1056 . . . 4 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → 𝑋𝐵)
86, 7ffvelrnd 6268 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐺𝑋) ∈ 𝐴)
9 fvco3 6185 . . 3 ((𝐺:𝐴𝐵 ∧ (𝐺𝑋) ∈ 𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹‘(𝐺‘(𝐺𝑋))))
102, 8, 9syl2anc 691 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹‘(𝐺‘(𝐺𝑋))))
11 f1ocnvfv2 6433 . . . 4 ((𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐺‘(𝐺𝑋)) = 𝑋)
12113adant1 1072 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐺‘(𝐺𝑋)) = 𝑋)
1312fveq2d 6107 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐹‘(𝐺‘(𝐺𝑋))) = (𝐹𝑋))
1410, 13eqtrd 2644 1 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ◡ccnv 5037   ∘ ccom 5042  Fun wfun 5798  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘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-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-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-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812 This theorem is referenced by:  f1ocan2fv  32692
 Copyright terms: Public domain W3C validator