Theorem fnun 5190

Description: The union of two functions with disjoint domains. (Contributed by set.mm contributors, 22-Sep-2004.)

Assertion

Ref	Expression
fnun	|- (((F Fn A /\ G Fn B) /\ (A i^i B) = (/)) -> (F u. G) Fn (A u. B))

Proof of Theorem fnun

Step	Hyp	Ref	Expression
1		df-fn 4791	. . 3 |- (F Fn A <-> (Fun F /\ dom F = A))
2		df-fn 4791	. . 3 |- (G Fn B <-> (Fun G /\ dom G = B))
3		ineq12 3453	. . . . . . . . . . 11 |- ((dom F = A /\ dom G = B) -> (dom F i^i dom G) = (A i^i B))
4	3	eqeq1d 2361	. . . . . . . . . 10 |- ((dom F = A /\ dom G = B) -> ((dom F i^i dom G) = (/) <-> (A i^i B) = (/)))
5	4	anbi2d 684	. . . . . . . . 9 |- ((dom F = A /\ dom G = B) -> (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) <-> ((Fun F /\ Fun G) /\ (A i^i B) = (/))))
6		funun 5147	. . . . . . . . 9 |- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> Fun (F u. G))
7	5, 6	syl6bir 220	. . . . . . . 8 |- ((dom F = A /\ dom G = B) -> (((Fun F /\ Fun G) /\ (A i^i B) = (/)) -> Fun (F u. G)))
8		dmun 4913	. . . . . . . . 9 |- dom (F u. G) = (dom F u. dom G)
9		uneq12 3414	. . . . . . . . 9 |- ((dom F = A /\ dom G = B) -> (dom F u. dom G) = (A u. B))
10	8, 9	syl5eq 2397	. . . . . . . 8 |- ((dom F = A /\ dom G = B) -> dom (F u. G) = (A u. B))
11	7, 10	jctird 528	. . . . . . 7 |- ((dom F = A /\ dom G = B) -> (((Fun F /\ Fun G) /\ (A i^i B) = (/)) -> (Fun (F u. G) /\ dom (F u. G) = (A u. B))))
12		df-fn 4791	. . . . . . 7 |- ((F u. G) Fn (A u. B) <-> (Fun (F u. G) /\ dom (F u. G) = (A u. B)))
13	11, 12	syl6ibr 218	. . . . . 6 |- ((dom F = A /\ dom G = B) -> (((Fun F /\ Fun G) /\ (A i^i B) = (/)) -> (F u. G) Fn (A u. B)))
14	13	exp3a 425	. . . . 5 |- ((dom F = A /\ dom G = B) -> ((Fun F /\ Fun G) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B))))
15	14	impcom 419	. . . 4 |- (((Fun F /\ Fun G) /\ (dom F = A /\ dom G = B)) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B)))
16	15	an4s 799	. . 3 |- (((Fun F /\ dom F = A) /\ (Fun G /\ dom G = B)) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B)))
17	1, 2, 16	syl2anb 465	. 2 |- ((F Fn A /\ G Fn B) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B)))
18	17	imp 418	1 |- (((F Fn A /\ G Fn B) /\ (A i^i B) = (/)) -> (F u. G) Fn (A u. B))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   u. cun 3208   i^i cin 3209  (/)c0 3551  dom cdm 4773  Fun wfun 4776   Fn wfn 4777

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-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791

This theorem is referenced by:  fnunsn  5191  fun  5237  f1oun  5305  fnfullfun  5859