Theorem fununfun 5934

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 5905	. . 3 |- ( Fun ( F u. G ) -> Rel ( F u. G ) )
2		relun 5235	. . 3 |- ( Rel ( F u. G ) <-> ( Rel F /\ Rel G ) )
3	1, 2	sylib 208	. 2 |- ( Fun ( F u. G ) -> ( Rel F /\ Rel G ) )
4		simpl 473	. . . . 5 |- ( ( Rel F /\ Rel G ) -> Rel F )
5		fununmo 5933	. . . . . 6 |- ( Fun ( F u. G ) -> E* y x F y )
6	5	alrimiv 1855	. . . . 5 |- ( Fun ( F u. G ) -> A. x E* y x F y )
7	4, 6	anim12i 590	. . . 4 |- ( ( ( Rel F /\ Rel G ) /\ Fun ( F u. G ) ) -> ( Rel F /\ A. x E* y x F y ) )
8		dffun6 5903	. . . 4 |- ( Fun F <-> ( Rel F /\ A. x E* y x F y ) )
9	7, 8	sylibr 224	. . 3 |- ( ( ( Rel F /\ Rel G ) /\ Fun ( F u. G ) ) -> Fun F )
10		simpr 477	. . . . 5 |- ( ( Rel F /\ Rel G ) -> Rel G )
11		uncom 3757	. . . . . . . 8 |- ( F u. G ) = ( G u. F )
12	11	funeqi 5909	. . . . . . 7 |- ( Fun ( F u. G ) <-> Fun ( G u. F ) )
13		fununmo 5933	. . . . . . 7 |- ( Fun ( G u. F ) -> E* y x G y )
14	12, 13	sylbi 207	. . . . . 6 |- ( Fun ( F u. G ) -> E* y x G y )
15	14	alrimiv 1855	. . . . 5 |- ( Fun ( F u. G ) -> A. x E* y x G y )
16	10, 15	anim12i 590	. . . 4 |- ( ( ( Rel F /\ Rel G ) /\ Fun ( F u. G ) ) -> ( Rel G /\ A. x E* y x G y ) )
17		dffun6 5903	. . . 4 |- ( Fun G <-> ( Rel G /\ A. x E* y x G y ) )
18	16, 17	sylibr 224	. . 3 |- ( ( ( Rel F /\ Rel G ) /\ Fun ( F u. G ) ) -> Fun G )
19	9, 18	jca 554	. 2 |- ( ( ( Rel F /\ Rel G ) /\ Fun ( F u. G ) ) -> ( Fun F /\ Fun G ) )
20	3, 19	mpancom 703	1 |- ( Fun ( F u. G ) -> ( Fun F /\ Fun G ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E*wmo 2471   u. cun 3572   class class class wbr 4653   Rel wrel 5119   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  fsuppunbi  8296