Theorem afv0nbfvbi 32475

Description: The function's value at an argument is an element of a set if and only if the value of the alternative function at this argument is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
afv0nbfvbi	|- ( (/) e/ B -> ( ( F''' A ) e. B <-> ( F ` A ) e. B ) )

Proof of Theorem afv0nbfvbi

Step	Hyp	Ref	Expression
1		afvvfveq 32472	. . 3 |- ( ( F''' A ) e. B -> ( F''' A ) = ( F ` A ) )
2		eleq1 2526	. . . 4 |- ( ( F''' A ) = ( F ` A ) -> ( ( F''' A ) e. B <-> ( F ` A ) e. B ) )
3	2	biimpd 207	. . 3 |- ( ( F''' A ) = ( F ` A ) -> ( ( F''' A ) e. B -> ( F ` A ) e. B ) )
4	1, 3	mpcom 36	. 2 |- ( ( F''' A ) e. B -> ( F ` A ) e. B )
5		elnelne2 2802	. . . . . 6 |- ( ( ( F ` A ) e. B /\ (/) e/ B ) -> ( F ` A ) =/= (/) )
6	5	ancoms 451	. . . . 5 |- ( ( (/) e/ B /\ ( F ` A ) e. B ) -> ( F ` A ) =/= (/) )
7		fvfundmfvn0 5880	. . . . 5 |- ( ( F ` A ) =/= (/) -> ( A e. dom F /\ Fun ( F |` { A } ) ) )
8		df-dfat 32440	. . . . . 6 |- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
9		afvfundmfveq 32462	. . . . . 6 |- ( F defAt A -> ( F''' A ) = ( F ` A ) )
10	8, 9	sylbir 213	. . . . 5 |- ( ( A e. dom F /\ Fun ( F |` { A } ) ) -> ( F''' A ) = ( F ` A ) )
11		eleq1 2526	. . . . . . 7 |- ( ( F ` A ) = ( F''' A ) -> ( ( F ` A ) e. B <-> ( F''' A ) e. B ) )
12	11	eqcoms 2466	. . . . . 6 |- ( ( F''' A ) = ( F ` A ) -> ( ( F ` A ) e. B <-> ( F''' A ) e. B ) )
13	12	biimpd 207	. . . . 5 |- ( ( F''' A ) = ( F ` A ) -> ( ( F ` A ) e. B -> ( F''' A ) e. B ) )
14	6, 7, 10, 13	4syl 21	. . . 4 |- ( ( (/) e/ B /\ ( F ` A ) e. B ) -> ( ( F ` A ) e. B -> ( F''' A ) e. B ) )
15	14	ex 432	. . 3 |- ( (/) e/ B -> ( ( F ` A ) e. B -> ( ( F ` A ) e. B -> ( F''' A ) e. B ) ) )
16	15	pm2.43d 48	. 2 |- ( (/) e/ B -> ( ( F ` A ) e. B -> ( F''' A ) e. B ) )
17	4, 16	impbid2 204	1 |- ( (/) e/ B -> ( ( F''' A ) e. B <-> ( F ` A ) e. B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   e/ wnel 2650   (/)c0 3783   {csn 4016   dom cdm 4988   |` cres 4990   Fun wfun 5564   ` cfv 5570   defAt wdfat 32437  '''cafv 32438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-res 5000  df-iota 5534  df-fun 5572  df-fv 5578  df-dfat 32440  df-afv 32441

This theorem is referenced by:  aov0nbovbi  32519