Theorem fununfun 5373

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 5343	. . 3 |- ( Fun ( F u. G ) -> Rel ( F u. G ) )
2		relun 4844	. . 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 5372	. . . . . 6 |- ( Fun ( F u. G ) -> E* y x F y )
6	5	alrimiv 1922	. . . . 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 5340	. . . 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 3351	. . . . . . . 8 |- ( F u. G ) = ( G u. F )
12	11	funeqi 5347	. . . . . . 7 |- ( Fun ( F u. G ) <-> Fun ( G u. F ) )
13		fununmo 5372	. . . . . . 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 1922	. . . . 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 5340	. . . 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 1395   E*wmo 2080   u. cun 3198   class class class wbr 4088   Rel wrel 4730   Fun wfun 5320

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-fun 5328

This theorem is referenced by: (None)