Theorem foun 5176

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 5139	. . . 4 |- ( F : A -onto-> B -> F Fn A )
2		fofn 5139	. . . 4 |- ( G : C -onto-> D -> G Fn C )
3	1, 2	anim12i 331	. . 3 |- ( ( F : A -onto-> B /\ G : C -onto-> D ) -> ( F Fn A /\ G Fn C ) )
4		fnun 5036	. . 3 |- ( ( ( F Fn A /\ G Fn C ) /\ ( A i^i C ) = (/) ) -> ( F u. G ) Fn ( A u. C ) )
5	3, 4	sylan 277	. 2 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ( F u. G ) Fn ( A u. C ) )
6		rnun 4762	. . 3 |- ran ( F u. G ) = ( ran F u. ran G )
7		forn 5140	. . . . 5 |- ( F : A -onto-> B -> ran F = B )
8	7	ad2antrr 472	. . . 4 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ran F = B )
9		forn 5140	. . . . 5 |- ( G : C -onto-> D -> ran G = D )
10	9	ad2antlr 473	. . . 4 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ran G = D )
11	8, 10	uneq12d 3128	. . 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 2126	. 2 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ran ( F u. G ) = ( B u. D ) )
13		df-fo 4938	. 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 408	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 102   = wceq 1285   u. cun 2972   i^i cin 2973   (/)c0 3258   ran crn 4372   Fn wfn 4927   -onto->wfo 4930

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-fun 4934  df-fn 4935  df-f 4936  df-fo 4938

This theorem is referenced by: (None)