Theorem afvfv0bi 38799

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 498	. . . 4 |- ( -. ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) <-> ( -. ( F''' A ) = (/) /\ -. ( F''' A ) = _V ) )
2		df-ne 2643	. . . . . . 7 |- ( ( F''' A ) =/= _V <-> -. ( F''' A ) = _V )
3		afvnufveq 38794	. . . . . . 7 |- ( ( F''' A ) =/= _V -> ( F''' A ) = ( F ` A ) )
4	2, 3	sylbir 218	. . . . . 6 |- ( -. ( F''' A ) = _V -> ( F''' A ) = ( F ` A ) )
5		eqeq1 2475	. . . . . . . 8 |- ( ( F''' A ) = ( F ` A ) -> ( ( F''' A ) = (/) <-> ( F ` A ) = (/) ) )
6	5	notbid 301	. . . . . . 7 |- ( ( F''' A ) = ( F ` A ) -> ( -. ( F''' A ) = (/) <-> -. ( F ` A ) = (/) ) )
7	6	biimpd 212	. . . . . 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 437	. . . 4 |- ( ( -. ( F''' A ) = (/) /\ -. ( F''' A ) = _V ) -> -. ( F ` A ) = (/) )
10	1, 9	sylbi 200	. . 3 |- ( -. ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) -> -. ( F ` A ) = (/) )
11	10	con4i 135	. 2 |- ( ( F ` A ) = (/) -> ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) )
12		afv0fv0 38796	. . 3 |- ( ( F''' A ) = (/) -> ( F ` A ) = (/) )
13		afvpcfv0 38793	. . 3 |- ( ( F''' A ) = _V -> ( F ` A ) = (/) )
14	12, 13	jaoi 386	. 2 |- ( ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) -> ( F ` A ) = (/) )
15	11, 14	impbii 192	1 |- ( ( F ` A ) = (/) <-> ( ( F''' A ) = (/) \/ ( F''' A ) = _V ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   = wceq 1452   =/= wne 2641   _Vcvv 3031   (/)c0 3722   ` cfv 5589  '''cafv 38760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-res 4851  df-iota 5553  df-fun 5591  df-fv 5597  df-dfat 38762  df-afv 38763

This theorem is referenced by:  aovov0bi  38843