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Theorem fvun 6307
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 5970 . . 3 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
2 funfv 6304 . . 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 5581 . . . 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 4478 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺) “ {𝐴}) = ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})))
7 uniun 4488 . . 3 ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ( (𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴}))
8 funfv 6304 . . . . . . 7 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
98eqcomd 2657 . . . . . 6 (Fun 𝐹 (𝐹 “ {𝐴}) = (𝐹𝐴))
10 funfv 6304 . . . . . . 7 (Fun 𝐺 → (𝐺𝐴) = (𝐺 “ {𝐴}))
1110eqcomd 2657 . . . . . 6 (Fun 𝐺 (𝐺 “ {𝐴}) = (𝐺𝐴))
129, 11anim12i 589 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺) → ( (𝐹 “ {𝐴}) = (𝐹𝐴) ∧ (𝐺 “ {𝐴}) = (𝐺𝐴)))
1312adantr 480 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ( (𝐹 “ {𝐴}) = (𝐹𝐴) ∧ (𝐺 “ {𝐴}) = (𝐺𝐴)))
14 uneq12 3795 . . . 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, 15syl5eq 2697 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ((𝐹𝐴) ∪ (𝐺𝐴)))
173, 6, 163eqtrd 2689 1 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺)‘𝐴) = ((𝐹𝐴) ∪ (𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  cun 3605  cin 3606  c0 3948  {csn 4210   cuni 4468  dom cdm 5143  cima 5146  Fun wfun 5920  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  fvun1  6308  undifixp  7986
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