Theorem afvfv0bi 38654

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 493	. . . 4 |- ( -. ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) <-> ( -. ( F''' A ) = (/) /\ -. ( F''' A ) = _V ) )
2		df-ne 2624	. . . . . . 7 |- ( ( F''' A ) =/= _V <-> -. ( F''' A ) = _V )
3		afvnufveq 38649	. . . . . . 7 |- ( ( F''' A ) =/= _V -> ( F''' A ) = ( F ` A ) )
4	2, 3	sylbir 217	. . . . . 6 |- ( -. ( F''' A ) = _V -> ( F''' A ) = ( F ` A ) )
5		eqeq1 2455	. . . . . . . 8 |- ( ( F''' A ) = ( F ` A ) -> ( ( F''' A ) = (/) <-> ( F ` A ) = (/) ) )
6	5	notbid 296	. . . . . . 7 |- ( ( F''' A ) = ( F ` A ) -> ( -. ( F''' A ) = (/) <-> -. ( F ` A ) = (/) ) )
7	6	biimpd 211	. . . . . 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 432	. . . 4 |- ( ( -. ( F''' A ) = (/) /\ -. ( F''' A ) = _V ) -> -. ( F ` A ) = (/) )
10	1, 9	sylbi 199	. . 3 |- ( -. ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) -> -. ( F ` A ) = (/) )
11	10	con4i 134	. 2 |- ( ( F ` A ) = (/) -> ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) )
12		afv0fv0 38651	. . 3 |- ( ( F''' A ) = (/) -> ( F ` A ) = (/) )
13		afvpcfv0 38648	. . 3 |- ( ( F''' A ) = _V -> ( F ` A ) = (/) )
14	12, 13	jaoi 381	. 2 |- ( ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) -> ( F ` A ) = (/) )
15	11, 14	impbii 191	1 |- ( ( F ` A ) = (/) <-> ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1444   =/= wne 2622   _Vcvv 3045   (/)c0 3731   ` cfv 5582  '''cafv 38615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-res 4846  df-iota 5546  df-fun 5584  df-fv 5590  df-dfat 38617  df-afv 38618

This theorem is referenced by:  aovov0bi  38698