Theorem fununfun 5363

Description: If the union of classes is a function, the classes itselves are functions. (Contributed by AV, 18-Jul-2019.)

Assertion

Ref	Expression
fununfun	|- ( Fun ( F u. G ) -> ( Fun F /\ Fun G ) )

Proof of Theorem fununfun

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funrel 5334	. . 3 |- ( Fun ( F u. G ) -> Rel ( F u. G ) )
2		relun 4835	. . 3 |- ( Rel ( F u. G ) <-> ( Rel F /\ Rel G ) )
3	1, 2	sylib 122	. 2 |- ( Fun ( F u. G ) -> ( Rel F /\ Rel G ) )
4		simpl 109	. . . . 5 |- ( ( Rel F /\ Rel G ) -> Rel F )
5		fununmo 5362	. . . . . 6 |- ( Fun ( F u. G ) -> E* y x F y )
6	5	alrimiv 1920	. . . . 5 |- ( Fun ( F u. G ) -> A. x E* y x F y )
7	4, 6	anim12i 338	. . . 4 |- ( ( ( Rel F /\ Rel G ) /\ Fun ( F u. G ) ) -> ( Rel F /\ A. x E* y x F y ) )
8		dffun6 5331	. . . 4 |- ( Fun F <-> ( Rel F /\ A. x E* y x F y ) )
9	7, 8	sylibr 134	. . 3 |- ( ( ( Rel F /\ Rel G ) /\ Fun ( F u. G ) ) -> Fun F )
10		simpr 110	. . . . 5 |- ( ( Rel F /\ Rel G ) -> Rel G )
11		uncom 3348	. . . . . . . 8 |- ( F u. G ) = ( G u. F )
12	11	funeqi 5338	. . . . . . 7 |- ( Fun ( F u. G ) <-> Fun ( G u. F ) )
13		fununmo 5362	. . . . . . 7 |- ( Fun ( G u. F ) -> E* y x G y )
14	12, 13	sylbi 121	. . . . . 6 |- ( Fun ( F u. G ) -> E* y x G y )
15	14	alrimiv 1920	. . . . 5 |- ( Fun ( F u. G ) -> A. x E* y x G y )
16	10, 15	anim12i 338	. . . 4 |- ( ( ( Rel F /\ Rel G ) /\ Fun ( F u. G ) ) -> ( Rel G /\ A. x E* y x G y ) )
17		dffun6 5331	. . . 4 |- ( Fun G <-> ( Rel G /\ A. x E* y x G y ) )
18	16, 17	sylibr 134	. . 3 |- ( ( ( Rel F /\ Rel G ) /\ Fun ( F u. G ) ) -> Fun G )
19	9, 18	jca 306	. 2 |- ( ( ( Rel F /\ Rel G ) /\ Fun ( F u. G ) ) -> ( Fun F /\ Fun G ) )
20	3, 19	mpancom 422	1 |- ( Fun ( F u. G ) -> ( Fun F /\ Fun G ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1393   E*wmo 2078   u. cun 3195   class class class wbr 4082   Rel wrel 4723   Fun wfun 5311

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-fun 5319

This theorem is referenced by: (None)