MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvun Structured version   Visualization version   GIF version

Theorem fvun 6982
Description: Value of the union of two functions when the domains are separate. (Contributed by FL, 7-Nov-2011.)
Assertion
Ref Expression
fvun (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺)‘𝐴) = ((𝐹𝐴) ∪ (𝐺𝐴)))

Proof of Theorem fvun
StepHypRef Expression
1 funun 6595 . . 3 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
2 funfv 6979 . . 3 (Fun (𝐹𝐺) → ((𝐹𝐺)‘𝐴) = ((𝐹𝐺) “ {𝐴}))
31, 2syl 17 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺)‘𝐴) = ((𝐹𝐺) “ {𝐴}))
4 imaundir 6151 . . . 4 ((𝐹𝐺) “ {𝐴}) = ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴}))
54a1i 11 . . 3 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺) “ {𝐴}) = ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})))
65unieqd 4923 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺) “ {𝐴}) = ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})))
7 uniun 4935 . . 3 ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ( (𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴}))
8 funfv 6979 . . . . . . 7 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
98eqcomd 2739 . . . . . 6 (Fun 𝐹 (𝐹 “ {𝐴}) = (𝐹𝐴))
10 funfv 6979 . . . . . . 7 (Fun 𝐺 → (𝐺𝐴) = (𝐺 “ {𝐴}))
1110eqcomd 2739 . . . . . 6 (Fun 𝐺 (𝐺 “ {𝐴}) = (𝐺𝐴))
129, 11anim12i 614 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺) → ( (𝐹 “ {𝐴}) = (𝐹𝐴) ∧ (𝐺 “ {𝐴}) = (𝐺𝐴)))
1312adantr 482 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ( (𝐹 “ {𝐴}) = (𝐹𝐴) ∧ (𝐺 “ {𝐴}) = (𝐺𝐴)))
14 uneq12 4159 . . . 4 (( (𝐹 “ {𝐴}) = (𝐹𝐴) ∧ (𝐺 “ {𝐴}) = (𝐺𝐴)) → ( (𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ((𝐹𝐴) ∪ (𝐺𝐴)))
1513, 14syl 17 . . 3 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ( (𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ((𝐹𝐴) ∪ (𝐺𝐴)))
167, 15eqtrid 2785 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ((𝐹𝐴) ∪ (𝐺𝐴)))
173, 6, 163eqtrd 2777 1 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺)‘𝐴) = ((𝐹𝐴) ∪ (𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  cun 3947  cin 3948  c0 4323  {csn 4629   cuni 4909  dom cdm 5677  cima 5680  Fun wfun 6538  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552
This theorem is referenced by:  fvun1  6983  undifixp  8928
  Copyright terms: Public domain W3C validator