Theorem afv0nbfvbi 30198

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 30195	. . 3 |- ( ( F''' A ) e. B -> ( F''' A ) = ( F ` A ) )
2		eleq1 2523	. . . 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		df-nel 2647	. . . . . . 7 |- ( (/) e/ B <-> -. (/) e. B )
6		nelne2 2778	. . . . . . 7 |- ( ( ( F ` A ) e. B /\ -. (/) e. B ) -> ( F ` A ) =/= (/) )
7	5, 6	sylan2b 475	. . . . . 6 |- ( ( ( F ` A ) e. B /\ (/) e/ B ) -> ( F ` A ) =/= (/) )
8	7	ancoms 453	. . . . 5 |- ( ( (/) e/ B /\ ( F ` A ) e. B ) -> ( F ` A ) =/= (/) )
9		fvfundmfvn0 5824	. . . . 5 |- ( ( F ` A ) =/= (/) -> ( A e. dom F /\ Fun ( F |` { A } ) ) )
10		df-dfat 30161	. . . . . 6 |- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
11		afvfundmfveq 30185	. . . . . 6 |- ( F defAt A -> ( F''' A ) = ( F ` A ) )
12	10, 11	sylbir 213	. . . . 5 |- ( ( A e. dom F /\ Fun ( F |` { A } ) ) -> ( F''' A ) = ( F ` A ) )
13		eleq1 2523	. . . . . . 7 |- ( ( F ` A ) = ( F''' A ) -> ( ( F ` A ) e. B <-> ( F''' A ) e. B ) )
14	13	eqcoms 2463	. . . . . 6 |- ( ( F''' A ) = ( F ` A ) -> ( ( F ` A ) e. B <-> ( F''' A ) e. B ) )
15	14	biimpd 207	. . . . 5 |- ( ( F''' A ) = ( F ` A ) -> ( ( F ` A ) e. B -> ( F''' A ) e. B ) )
16	8, 9, 12, 15	4syl 21	. . . 4 |- ( ( (/) e/ B /\ ( F ` A ) e. B ) -> ( ( F ` A ) e. B -> ( F''' A ) e. B ) )
17	16	ex 434	. . 3 |- ( (/) e/ B -> ( ( F ` A ) e. B -> ( ( F ` A ) e. B -> ( F''' A ) e. B ) ) )
18	17	pm2.43d 48	. 2 |- ( (/) e/ B -> ( ( F ` A ) e. B -> ( F''' A ) e. B ) )
19	4, 18	impbid2 204	1 |- ( (/) e/ B -> ( ( F''' A ) e. B <-> ( F ` A ) e. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2644   e/ wnel 2645   (/)c0 3738   {csn 3978   dom cdm 4941   |` cres 4943   Fun wfun 5513   ` cfv 5519   defAt wdfat 30158  '''cafv 30159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-res 4953  df-iota 5482  df-fun 5521  df-fv 5527  df-dfat 30161  df-afv 30162

This theorem is referenced by:  aov0nbovbi  30242