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Theorem fvun 6225
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 5890 . . 3 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
2 funfv 6222 . . 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 5505 . . . 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 4412 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺) “ {𝐴}) = ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})))
7 uniun 4422 . . 3 ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ( (𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴}))
8 funfv 6222 . . . . . . 7 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
98eqcomd 2627 . . . . . 6 (Fun 𝐹 (𝐹 “ {𝐴}) = (𝐹𝐴))
10 funfv 6222 . . . . . . 7 (Fun 𝐺 → (𝐺𝐴) = (𝐺 “ {𝐴}))
1110eqcomd 2627 . . . . . 6 (Fun 𝐺 (𝐺 “ {𝐴}) = (𝐺𝐴))
129, 11anim12i 589 . . . . 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 3740 . . . 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 2667 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ((𝐹𝐴) ∪ (𝐺𝐴)))
173, 6, 163eqtrd 2659 1 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺)‘𝐴) = ((𝐹𝐴) ∪ (𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  cun 3553  cin 3554  c0 3891  {csn 4148   cuni 4402  dom cdm 5074  cima 5077  Fun wfun 5841  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855
This theorem is referenced by:  fvun1  6226  undifixp  7888
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