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Theorem fvun 6491
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 6144 . . 3 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
2 funfv 6488 . . 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 5761 . . . 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 4636 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺) “ {𝐴}) = ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})))
7 uniun 4647 . . 3 ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ( (𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴}))
8 funfv 6488 . . . . . . 7 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
98eqcomd 2803 . . . . . 6 (Fun 𝐹 (𝐹 “ {𝐴}) = (𝐹𝐴))
10 funfv 6488 . . . . . . 7 (Fun 𝐺 → (𝐺𝐴) = (𝐺 “ {𝐴}))
1110eqcomd 2803 . . . . . 6 (Fun 𝐺 (𝐺 “ {𝐴}) = (𝐺𝐴))
129, 11anim12i 607 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺) → ( (𝐹 “ {𝐴}) = (𝐹𝐴) ∧ (𝐺 “ {𝐴}) = (𝐺𝐴)))
1312adantr 473 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ( (𝐹 “ {𝐴}) = (𝐹𝐴) ∧ (𝐺 “ {𝐴}) = (𝐺𝐴)))
14 uneq12 3958 . . . 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 2843 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ((𝐹𝐴) ∪ (𝐺𝐴)))
173, 6, 163eqtrd 2835 1 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺)‘𝐴) = ((𝐹𝐴) ∪ (𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  cun 3765  cin 3766  c0 4113  {csn 4366   cuni 4626  dom cdm 5310  cima 5313  Fun wfun 6093  cfv 6099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-fv 6107
This theorem is referenced by:  fvun1  6492  undifixp  8182
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