Theorem foun 5287

Description: The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)

Assertion

Ref	Expression
foun	|- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ( F u. G ) : ( A u. C ) -onto-> ( B u. D ) )

Proof of Theorem foun

Step	Hyp	Ref	Expression
1		fofn 5250	. . . 4 |- ( F : A -onto-> B -> F Fn A )
2		fofn 5250	. . . 4 |- ( G : C -onto-> D -> G Fn C )
3	1, 2	anim12i 332	. . 3 |- ( ( F : A -onto-> B /\ G : C -onto-> D ) -> ( F Fn A /\ G Fn C ) )
4		fnun 5135	. . 3 |- ( ( ( F Fn A /\ G Fn C ) /\ ( A i^i C ) = (/) ) -> ( F u. G ) Fn ( A u. C ) )
5	3, 4	sylan 278	. 2 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ( F u. G ) Fn ( A u. C ) )
6		rnun 4855	. . 3 |- ran ( F u. G ) = ( ran F u. ran G )
7		forn 5251	. . . . 5 |- ( F : A -onto-> B -> ran F = B )
8	7	ad2antrr 473	. . . 4 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ran F = B )
9		forn 5251	. . . . 5 |- ( G : C -onto-> D -> ran G = D )
10	9	ad2antlr 474	. . . 4 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ran G = D )
11	8, 10	uneq12d 3158	. . 3 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ( ran F u. ran G ) = ( B u. D ) )
12	6, 11	syl5eq 2133	. 2 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ran ( F u. G ) = ( B u. D ) )
13		df-fo 5036	. 2 |- ( ( F u. G ) : ( A u. C ) -onto-> ( B u. D ) <-> ( ( F u. G ) Fn ( A u. C ) /\ ran ( F u. G ) = ( B u. D ) ) )
14	5, 12, 13	sylanbrc 409	1 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ( F u. G ) : ( A u. C ) -onto-> ( B u. D ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   u. cun 3000   i^i cin 3001   (/)c0 3289   ran crn 4455   Fn wfn 5025   -onto->wfo 5028

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2624  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-id 4131  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-fun 5032  df-fn 5033  df-f 5034  df-fo 5036

This theorem is referenced by: (None)