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) = ∅) → ((F ∪ G) ‘A) = ((F ‘A) ∪ (G ‘A)))

Proof of Theorem fvun

Step	Hyp	Ref	Expression
1		funun 5146	. . 3 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → Fun (F ∪ G))
2		funfv 5375	. . 3 ⊢ (Fun (F ∪ G) → ((F ∪ G) ‘A) = ∪((F ∪ G) “ {A}))
3	1, 2	syl 15	. 2 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → ((F ∪ G) ‘A) = ∪((F ∪ G) “ {A}))
4		imaundir 5040	. . . 4 ⊢ ((F ∪ G) “ {A}) = ((F “ {A}) ∪ (G “ {A}))
5	4	a1i 10	. . 3 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → ((F ∪ G) “ {A}) = ((F “ {A}) ∪ (G “ {A})))
6	5	unieqd 3902	. 2 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → ∪((F ∪ G) “ {A}) = ∪((F “ {A}) ∪ (G “ {A})))
7		uniun 3910	. . 3 ⊢ ∪((F “ {A}) ∪ (G “ {A})) = (∪(F “ {A}) ∪ ∪(G “ {A}))
8		funfv 5375	. . . . . . 7 ⊢ (Fun F → (F ‘A) = ∪(F “ {A}))
9	8	eqcomd 2358	. . . . . 6 ⊢ (Fun F → ∪(F “ {A}) = (F ‘A))
10		funfv 5375	. . . . . . 7 ⊢ (Fun G → (G ‘A) = ∪(G “ {A}))
11	10	eqcomd 2358	. . . . . 6 ⊢ (Fun G → ∪(G “ {A}) = (G ‘A))
12	9, 11	anim12i 549	. . . . 5 ⊢ ((Fun F ∧ Fun G) → (∪(F “ {A}) = (F ‘A) ∧ ∪(G “ {A}) = (G ‘A)))
13	12	adantr 451	. . . 4 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → (∪(F “ {A}) = (F ‘A) ∧ ∪(G “ {A}) = (G ‘A)))
14		uneq12 3413	. . . 4 ⊢ ((∪(F “ {A}) = (F ‘A) ∧ ∪(G “ {A}) = (G ‘A)) → (∪(F “ {A}) ∪ ∪(G “ {A})) = ((F ‘A) ∪ (G ‘A)))
15	13, 14	syl 15	. . 3 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → (∪(F “ {A}) ∪ ∪(G “ {A})) = ((F ‘A) ∪ (G ‘A)))
16	7, 15	syl5eq 2397	. 2 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → ∪((F “ {A}) ∪ (G “ {A})) = ((F ‘A) ∪ (G ‘A)))
17	3, 6, 16	3eqtrd 2389	1 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → ((F ∪ G) ‘A) = ((F ‘A) ∪ (G ‘A)))

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