Theorem fununfun 5419

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 5389	. . 3 |- ( Fun ( F u. G ) -> Rel ( F u. G ) )
2		relun 4889	. . 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 5418	. . . . . 6 |- ( Fun ( F u. G ) -> E* y x F y )
6	5	alrimiv 1927	. . . . 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 5386	. . . 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 3373	. . . . . . . 8 |- ( F u. G ) = ( G u. F )
12	11	funeqi 5393	. . . . . . 7 |- ( Fun ( F u. G ) <-> Fun ( G u. F ) )
13		fununmo 5418	. . . . . . 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 1927	. . . . 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 5386	. . . 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 426	1 |- ( Fun ( F u. G ) -> ( Fun F /\ Fun G ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1400   E*wmo 2087   u. cun 3218   class class class wbr 4125   Rel wrel 4774   Fun wfun 5366

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 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-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-fun 5374

This theorem is referenced by: (None)