Theorem afvfv0bi 41232

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 511	. . . 4 |- ( -. ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) <-> ( -. ( F''' A ) = (/) /\ -. ( F''' A ) = _V ) )
2		df-ne 2795	. . . . . . 7 |- ( ( F''' A ) =/= _V <-> -. ( F''' A ) = _V )
3		afvnufveq 41227	. . . . . . 7 |- ( ( F''' A ) =/= _V -> ( F''' A ) = ( F ` A ) )
4	2, 3	sylbir 225	. . . . . 6 |- ( -. ( F''' A ) = _V -> ( F''' A ) = ( F ` A ) )
5		eqeq1 2626	. . . . . . . 8 |- ( ( F''' A ) = ( F ` A ) -> ( ( F''' A ) = (/) <-> ( F ` A ) = (/) ) )
6	5	notbid 308	. . . . . . 7 |- ( ( F''' A ) = ( F ` A ) -> ( -. ( F''' A ) = (/) <-> -. ( F ` A ) = (/) ) )
7	6	biimpd 219	. . . . . 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 446	. . . 4 |- ( ( -. ( F''' A ) = (/) /\ -. ( F''' A ) = _V ) -> -. ( F ` A ) = (/) )
10	1, 9	sylbi 207	. . 3 |- ( -. ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) -> -. ( F ` A ) = (/) )
11	10	con4i 113	. 2 |- ( ( F ` A ) = (/) -> ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) )
12		afv0fv0 41229	. . 3 |- ( ( F''' A ) = (/) -> ( F ` A ) = (/) )
13		afvpcfv0 41226	. . 3 |- ( ( F''' A ) = _V -> ( F ` A ) = (/) )
14	12, 13	jaoi 394	. 2 |- ( ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) -> ( F ` A ) = (/) )
15	11, 14	impbii 199	1 |- ( ( F ` A ) = (/) <-> ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   =/= wne 2794   _Vcvv 3200   (/)c0 3915   ` cfv 5888  '''cafv 41194

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-8 1992  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-pow 4843  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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-dfat 41196  df-afv 41197

This theorem is referenced by:  aovov0bi  41276