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Theorem fvun 5378
 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 5146 . . 3 (((Fun F Fun G) (dom F ∩ dom G) = ) → Fun (FG))
2 funfv 5375 . . 3 (Fun (FG) → ((FG) ‘A) = ((FG) “ {A}))
31, 2syl 15 . 2 (((Fun F Fun G) (dom F ∩ dom G) = ) → ((FG) ‘A) = ((FG) “ {A}))
4 imaundir 5040 . . . 4 ((FG) “ {A}) = ((F “ {A}) ∪ (G “ {A}))
54a1i 10 . . 3 (((Fun F Fun G) (dom F ∩ dom G) = ) → ((FG) “ {A}) = ((F “ {A}) ∪ (G “ {A})))
65unieqd 3902 . 2 (((Fun F Fun G) (dom F ∩ dom G) = ) → ((FG) “ {A}) = ((F “ {A}) ∪ (G “ {A})))
7 uniun 3910 . . 3 ((F “ {A}) ∪ (G “ {A})) = ((F “ {A}) ∪ (G “ {A}))
8 funfv 5375 . . . . . . 7 (Fun F → (FA) = (F “ {A}))
98eqcomd 2358 . . . . . 6 (Fun F(F “ {A}) = (FA))
10 funfv 5375 . . . . . . 7 (Fun G → (GA) = (G “ {A}))
1110eqcomd 2358 . . . . . 6 (Fun G(G “ {A}) = (GA))
129, 11anim12i 549 . . . . 5 ((Fun F Fun G) → ((F “ {A}) = (FA) (G “ {A}) = (GA)))
1312adantr 451 . . . 4 (((Fun F Fun G) (dom F ∩ dom G) = ) → ((F “ {A}) = (FA) (G “ {A}) = (GA)))
14 uneq12 3413 . . . 4 (((F “ {A}) = (FA) (G “ {A}) = (GA)) → ((F “ {A}) ∪ (G “ {A})) = ((FA) ∪ (GA)))
1513, 14syl 15 . . 3 (((Fun F Fun G) (dom F ∩ dom G) = ) → ((F “ {A}) ∪ (G “ {A})) = ((FA) ∪ (GA)))
167, 15syl5eq 2397 . 2 (((Fun F Fun G) (dom F ∩ dom G) = ) → ((F “ {A}) ∪ (G “ {A})) = ((FA) ∪ (GA)))
173, 6, 163eqtrd 2389 1 (((Fun F Fun G) (dom F ∩ dom G) = ) → ((FG) ‘A) = ((FA) ∪ (GA)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  ∪cuni 3891   “ cima 4722  dom cdm 4772  Fun wfun 4775   ‘cfv 4781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-fv 4795 This theorem is referenced by:  fvun1  5379
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