Theorem afv0nbfvbi 38691

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 38688	. . 3 |- ( ( F''' A ) e. B -> ( F''' A ) = ( F ` A ) )
2		eleq1 2528	. . . 4 |- ( ( F''' A ) = ( F ` A ) -> ( ( F''' A ) e. B <-> ( F ` A ) e. B ) )
3	2	biimpd 212	. . 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 2747	. . . . . 6 |- ( ( ( F ` A ) e. B /\ (/) e/ B ) -> ( F ` A ) =/= (/) )
6	5	ancoms 459	. . . . 5 |- ( ( (/) e/ B /\ ( F ` A ) e. B ) -> ( F ` A ) =/= (/) )
7		fvfundmfvn0 5920	. . . . 5 |- ( ( F ` A ) =/= (/) -> ( A e. dom F /\ Fun ( F |` { A } ) ) )
8		df-dfat 38655	. . . . . 6 |- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
9		afvfundmfveq 38678	. . . . . 6 |- ( F defAt A -> ( F''' A ) = ( F ` A ) )
10	8, 9	sylbir 218	. . . . 5 |- ( ( A e. dom F /\ Fun ( F |` { A } ) ) -> ( F''' A ) = ( F ` A ) )
11		eleq1 2528	. . . . . . 7 |- ( ( F ` A ) = ( F''' A ) -> ( ( F ` A ) e. B <-> ( F''' A ) e. B ) )
12	11	eqcoms 2470	. . . . . 6 |- ( ( F''' A ) = ( F ` A ) -> ( ( F ` A ) e. B <-> ( F''' A ) e. B ) )
13	12	biimpd 212	. . . . 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 440	. . 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 209	1 |- ( (/) e/ B -> ( ( F''' A ) e. B <-> ( F ` A ) e. B ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   =/= wne 2633   e/ wnel 2634   (/)c0 3743   {csn 3980   dom cdm 4853   |` cres 4855   Fun wfun 5595   ` cfv 5601   defAt wdfat 38652  '''cafv 38653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-res 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-dfat 38655  df-afv 38656

This theorem is referenced by:  aov0nbovbi  38735