Theorem fvun 5379

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 5147	. . 3 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → Fun (F ∪ G))
2		funfv 5376	. . 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 5041	. . . 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 3903	. 2 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → ∪((F ∪ G) “ {A}) = ∪((F “ {A}) ∪ (G “ {A})))
7		uniun 3911	. . 3 ⊢ ∪((F “ {A}) ∪ (G “ {A})) = (∪(F “ {A}) ∪ ∪(G “ {A}))
8		funfv 5376	. . . . . . 7 ⊢ (Fun F → (F ‘A) = ∪(F “ {A}))
9	8	eqcomd 2358	. . . . . 6 ⊢ (Fun F → ∪(F “ {A}) = (F ‘A))
10		funfv 5376	. . . . . . 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 3414	. . . 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 3208   ∩ cin 3209  ∅c0 3551  {csn 3738  ∪cuni 3892   “ cima 4723  dom cdm 4773  Fun wfun 4776   ‘cfv 4782

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-fv 4796

This theorem is referenced by:  fvun1  5380