Theorem afvfv0bi 38043

Description: The function's value at an argument is the empty set if and only if the value of the alternative function at this argument is either the empty set or the universe. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
afvfv0bi	|- ( ( F ` A ) = (/) <-> ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) )

Proof of Theorem afvfv0bi

Step	Hyp	Ref	Expression
1		ioran 492	. . . 4 |- ( -. ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) <-> ( -. ( F''' A ) = (/) /\ -. ( F''' A ) = _V ) )
2		df-ne 2627	. . . . . . 7 |- ( ( F''' A ) =/= _V <-> -. ( F''' A ) = _V )
3		afvnufveq 38038	. . . . . . 7 |- ( ( F''' A ) =/= _V -> ( F''' A ) = ( F ` A ) )
4	2, 3	sylbir 216	. . . . . 6 |- ( -. ( F''' A ) = _V -> ( F''' A ) = ( F ` A ) )
5		eqeq1 2433	. . . . . . . 8 |- ( ( F''' A ) = ( F ` A ) -> ( ( F''' A ) = (/) <-> ( F ` A ) = (/) ) )
6	5	notbid 295	. . . . . . 7 |- ( ( F''' A ) = ( F ` A ) -> ( -. ( F''' A ) = (/) <-> -. ( F ` A ) = (/) ) )
7	6	biimpd 210	. . . . . 6 |- ( ( F''' A ) = ( F ` A ) -> ( -. ( F''' A ) = (/) -> -. ( F ` A ) = (/) ) )
8	4, 7	syl 17	. . . . 5 |- ( -. ( F''' A ) = _V -> ( -. ( F''' A ) = (/) -> -. ( F ` A ) = (/) ) )
9	8	impcom 431	. . . 4 |- ( ( -. ( F''' A ) = (/) /\ -. ( F''' A ) = _V ) -> -. ( F ` A ) = (/) )
10	1, 9	sylbi 198	. . 3 |- ( -. ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) -> -. ( F ` A ) = (/) )
11	10	con4i 133	. 2 |- ( ( F ` A ) = (/) -> ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) )
12		afv0fv0 38040	. . 3 |- ( ( F''' A ) = (/) -> ( F ` A ) = (/) )
13		afvpcfv0 38037	. . 3 |- ( ( F''' A ) = _V -> ( F ` A ) = (/) )
14	12, 13	jaoi 380	. 2 |- ( ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) -> ( F ` A ) = (/) )
15	11, 14	impbii 190	1 |- ( ( F ` A ) = (/) <-> ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   \/ wo 369   /\ wa 370   = wceq 1437   =/= wne 2625   _Vcvv 3087   (/)c0 3767   ` cfv 5601  '''cafv 38005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-res 4866  df-iota 5565  df-fun 5603  df-fv 5609  df-dfat 38007  df-afv 38008

This theorem is referenced by:  aovov0bi  38087