Theorem afvpcfv0 41226

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

Assertion

Ref	Expression
afvpcfv0	|- ( ( F''' A ) = _V -> ( F ` A ) = (/) )

Proof of Theorem afvpcfv0

Step	Hyp	Ref	Expression
1		dfafv2 41212	. . 3 |- ( F''' A ) = if ( F defAt A , ( F ` A ) , _V )
2	1	eqeq1i 2627	. 2 |- ( ( F''' A ) = _V <-> if ( F defAt A , ( F ` A ) , _V ) = _V )
3		eqcom 2629	. . . 4 |- ( if ( F defAt A , ( F ` A ) , _V ) = _V <-> _V = if ( F defAt A , ( F ` A ) , _V ) )
4		eqif 4126	. . . 4 |- ( _V = if ( F defAt A , ( F ` A ) , _V ) <-> ( ( F defAt A /\ _V = ( F ` A ) ) \/ ( -. F defAt A /\ _V = _V ) ) )
5	3, 4	bitri 264	. . 3 |- ( if ( F defAt A , ( F ` A ) , _V ) = _V <-> ( ( F defAt A /\ _V = ( F ` A ) ) \/ ( -. F defAt A /\ _V = _V ) ) )
6		fveqvfvv 41204	. . . . . 6 |- ( ( F ` A ) = _V -> ( F ` A ) = (/) )
7	6	eqcoms 2630	. . . . 5 |- ( _V = ( F ` A ) -> ( F ` A ) = (/) )
8	7	adantl 482	. . . 4 |- ( ( F defAt A /\ _V = ( F ` A ) ) -> ( F ` A ) = (/) )
9		fvfundmfvn0 6226	. . . . . . 7 |- ( ( F ` A ) =/= (/) -> ( A e. dom F /\ Fun ( F |` { A } ) ) )
10		df-dfat 41196	. . . . . . 7 |- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
11	9, 10	sylibr 224	. . . . . 6 |- ( ( F ` A ) =/= (/) -> F defAt A )
12	11	necon1bi 2822	. . . . 5 |- ( -. F defAt A -> ( F ` A ) = (/) )
13	12	adantr 481	. . . 4 |- ( ( -. F defAt A /\ _V = _V ) -> ( F ` A ) = (/) )
14	8, 13	jaoi 394	. . 3 |- ( ( ( F defAt A /\ _V = ( F ` A ) ) \/ ( -. F defAt A /\ _V = _V ) ) -> ( F ` A ) = (/) )
15	5, 14	sylbi 207	. 2 |- ( if ( F defAt A , ( F ` A ) , _V ) = _V -> ( F ` A ) = (/) )
16	2, 15	sylbi 207	1 |- ( ( F''' A ) = _V -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   ifcif 4086   {csn 4177   dom cdm 5114   |` cres 5116   Fun wfun 5882   ` cfv 5888   defAt wdfat 41193  '''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:  afvfv0bi  41232  aovpcov0  41270