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Theorem fvun 6750
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 6397 . . 3 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
2 funfv 6747 . . 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 6007 . . . 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 4847 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺) “ {𝐴}) = ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})))
7 uniun 4856 . . 3 ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ( (𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴}))
8 funfv 6747 . . . . . . 7 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
98eqcomd 2832 . . . . . 6 (Fun 𝐹 (𝐹 “ {𝐴}) = (𝐹𝐴))
10 funfv 6747 . . . . . . 7 (Fun 𝐺 → (𝐺𝐴) = (𝐺 “ {𝐴}))
1110eqcomd 2832 . . . . . 6 (Fun 𝐺 (𝐺 “ {𝐴}) = (𝐺𝐴))
129, 11anim12i 612 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺) → ( (𝐹 “ {𝐴}) = (𝐹𝐴) ∧ (𝐺 “ {𝐴}) = (𝐺𝐴)))
1312adantr 481 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ( (𝐹 “ {𝐴}) = (𝐹𝐴) ∧ (𝐺 “ {𝐴}) = (𝐺𝐴)))
14 uneq12 4138 . . . 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 2873 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ((𝐹𝐴) ∪ (𝐺𝐴)))
173, 6, 163eqtrd 2865 1 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺)‘𝐴) = ((𝐹𝐴) ∪ (𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  cun 3938  cin 3939  c0 4295  {csn 4564   cuni 4837  dom cdm 5554  cima 5557  Fun wfun 6346  cfv 6352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-fv 6360
This theorem is referenced by:  fvun1  6751  undifixp  8487
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