Theorem fununmo 5403

Description: If the union of classes is a function, there is at most one element in relation to an arbitrary element regarding one of these classes. (Contributed by AV, 18-Jul-2019.)

Assertion

Ref	Expression
fununmo	|- ( Fun ( F u. G ) -> E* y x F y )

Distinct variable groups:   x, y   y, F   y, G

Allowed substitution hints:   F( x)   G( x)

Proof of Theorem fununmo

Step	Hyp	Ref	Expression
1		funmo 5372	. 2 |- ( Fun ( F u. G ) -> E* y x ( F u. G ) y )
2		orc 720	. . . 4 |- ( x F y -> ( x F y \/ x G y ) )
3		brun 4166	. . . 4 |- ( x ( F u. G ) y <-> ( x F y \/ x G y ) )
4	2, 3	sylibr 134	. . 3 |- ( x F y -> x ( F u. G ) y )
5	4	moimi 2148	. 2 |- ( E* y x ( F u. G ) y -> E* y x F y )
6	1, 5	syl 14	1 |- ( Fun ( F u. G ) -> E* y x F y )

Syntax hints:   -> wi 4   \/ wo 716   E*wmo 2083   u. cun 3212   class class class wbr 4114   Fun wfun 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-fun 5359

This theorem is referenced by:  fununfun  5404