Theorem afvfv0bi 31720

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 490	. . . 4 |- ( -. ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) <-> ( -. ( F''' A ) = (/) /\ -. ( F''' A ) = _V ) )
2		df-ne 2664	. . . . . . 7 |- ( ( F''' A ) =/= _V <-> -. ( F''' A ) = _V )
3		afvnufveq 31715	. . . . . . 7 |- ( ( F''' A ) =/= _V -> ( F''' A ) = ( F ` A ) )
4	2, 3	sylbir 213	. . . . . 6 |- ( -. ( F''' A ) = _V -> ( F''' A ) = ( F ` A ) )
5		eqeq1 2471	. . . . . . . 8 |- ( ( F''' A ) = ( F ` A ) -> ( ( F''' A ) = (/) <-> ( F ` A ) = (/) ) )
6	5	notbid 294	. . . . . . 7 |- ( ( F''' A ) = ( F ` A ) -> ( -. ( F''' A ) = (/) <-> -. ( F ` A ) = (/) ) )
7	6	biimpd 207	. . . . . 6 |- ( ( F''' A ) = ( F ` A ) -> ( -. ( F''' A ) = (/) -> -. ( F ` A ) = (/) ) )
8	4, 7	syl 16	. . . . 5 |- ( -. ( F''' A ) = _V -> ( -. ( F''' A ) = (/) -> -. ( F ` A ) = (/) ) )
9	8	impcom 430	. . . 4 |- ( ( -. ( F''' A ) = (/) /\ -. ( F''' A ) = _V ) -> -. ( F ` A ) = (/) )
10	1, 9	sylbi 195	. . 3 |- ( -. ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) -> -. ( F ` A ) = (/) )
11	10	con4i 130	. 2 |- ( ( F ` A ) = (/) -> ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) )
12		afv0fv0 31717	. . 3 |- ( ( F''' A ) = (/) -> ( F ` A ) = (/) )
13		afvpcfv0 31714	. . 3 |- ( ( F''' A ) = _V -> ( F ` A ) = (/) )
14	12, 13	jaoi 379	. 2 |- ( ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) -> ( F ` A ) = (/) )
15	11, 14	impbii 188	1 |- ( ( F ` A ) = (/) <-> ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1379   =/= wne 2662   _Vcvv 3113   (/)c0 3785   ` cfv 5587  '''cafv 31682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5550  df-fun 5589  df-fv 5595  df-dfat 31684  df-afv 31685

This theorem is referenced by:  aovov0bi  31764