Theorem afv0nbfvbi 38043

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 38040	. . 3 |- ( ( F''' A ) e. B -> ( F''' A ) = ( F ` A ) )
2		eleq1 2501	. . . 4 |- ( ( F''' A ) = ( F ` A ) -> ( ( F''' A ) e. B <-> ( F ` A ) e. B ) )
3	2	biimpd 210	. . 3 |- ( ( F''' A ) = ( F ` A ) -> ( ( F''' A ) e. B -> ( F ` A ) e. B ) )
4	1, 3	mpcom 37	. 2 |- ( ( F''' A ) e. B -> ( F ` A ) e. B )
5		elnelne2 2780	. . . . . 6 |- ( ( ( F ` A ) e. B /\ (/) e/ B ) -> ( F ` A ) =/= (/) )
6	5	ancoms 454	. . . . 5 |- ( ( (/) e/ B /\ ( F ` A ) e. B ) -> ( F ` A ) =/= (/) )
7		fvfundmfvn0 5913	. . . . 5 |- ( ( F ` A ) =/= (/) -> ( A e. dom F /\ Fun ( F |` { A } ) ) )
8		df-dfat 38008	. . . . . 6 |- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
9		afvfundmfveq 38030	. . . . . 6 |- ( F defAt A -> ( F''' A ) = ( F ` A ) )
10	8, 9	sylbir 216	. . . . 5 |- ( ( A e. dom F /\ Fun ( F |` { A } ) ) -> ( F''' A ) = ( F ` A ) )
11		eleq1 2501	. . . . . . 7 |- ( ( F ` A ) = ( F''' A ) -> ( ( F ` A ) e. B <-> ( F''' A ) e. B ) )
12	11	eqcoms 2441	. . . . . 6 |- ( ( F''' A ) = ( F ` A ) -> ( ( F ` A ) e. B <-> ( F''' A ) e. B ) )
13	12	biimpd 210	. . . . 5 |- ( ( F''' A ) = ( F ` A ) -> ( ( F ` A ) e. B -> ( F''' A ) e. B ) )
14	6, 7, 10, 13	4syl 19	. . . 4 |- ( ( (/) e/ B /\ ( F ` A ) e. B ) -> ( ( F ` A ) e. B -> ( F''' A ) e. B ) )
15	14	ex 435	. . 3 |- ( (/) e/ B -> ( ( F ` A ) e. B -> ( ( F ` A ) e. B -> ( F''' A ) e. B ) ) )
16	15	pm2.43d 50	. 2 |- ( (/) e/ B -> ( ( F ` A ) e. B -> ( F''' A ) e. B ) )
17	4, 16	impbid2 207	1 |- ( (/) e/ B -> ( ( F''' A ) e. B <-> ( F ` A ) e. B ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1870   =/= wne 2625   e/ wnel 2626   (/)c0 3767   {csn 4002   dom cdm 4854   |` cres 4856   Fun wfun 5595   ` cfv 5601   defAt wdfat 38005  '''cafv 38006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-res 4866  df-iota 5565  df-fun 5603  df-fv 5609  df-dfat 38008  df-afv 38009

This theorem is referenced by:  aov0nbovbi  38087