Theorem foun 5653

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 5612	. . . 4 |- ( F : A -onto-> B -> F Fn A )
2		fofn 5612	. . . 4 |- ( G : C -onto-> D -> G Fn C )
3	1, 2	anim12i 338	. . 3 |- ( ( F : A -onto-> B /\ G : C -onto-> D ) -> ( F Fn A /\ G Fn C ) )
4		fnun 5484	. . 3 |- ( ( ( F Fn A /\ G Fn C ) /\ ( A i^i C ) = (/) ) -> ( F u. G ) Fn ( A u. C ) )
5	3, 4	sylan 283	. 2 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ( F u. G ) Fn ( A u. C ) )
6		rnun 5191	. . 3 |- ran ( F u. G ) = ( ran F u. ran G )
7		forn 5613	. . . . 5 |- ( F : A -onto-> B -> ran F = B )
8	7	ad2antrr 492	. . . 4 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ran F = B )
9		forn 5613	. . . . 5 |- ( G : C -onto-> D -> ran G = D )
10	9	ad2antlr 493	. . . 4 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ran G = D )
11	8, 10	uneq12d 3384	. . 3 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ( ran F u. ran G ) = ( B u. D ) )
12	6, 11	eqtrid 2283	. 2 |- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ran ( F u. G ) = ( B u. D ) )
13		df-fo 5378	. 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 421	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 104   = wceq 1402   u. cun 3218   i^i cin 3219   (/)c0 3520   ran crn 4770   Fn wfn 5367   -onto->wfo 5370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378

This theorem is referenced by: (None)