Theorem suppssfv 6103

Description: Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Hypotheses

Ref	Expression
suppssfv.a	|- ( ph -> ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) C_ L )
suppssfv.f	|- ( ph -> ( F ` Y ) = Z )
suppssfv.v	|- ( ( ph /\ x e. D ) -> A e. V )

Assertion

Ref	Expression
suppssfv	|- ( ph -> ( `' ( x e. D |-> ( F ` A ) ) " ( _V \ { Z } ) ) C_ L )

Distinct variable groups:   ph, x   x, Y   x, Z

Allowed substitution hints:   A( x)   D( x)   F( x)   L( x)   V( x)

Proof of Theorem suppssfv

Step	Hyp	Ref	Expression
1		eldifsni 3736	. . . . 5 |- ( ( F ` A ) e. ( _V \ { Z } ) -> ( F ` A ) =/= Z )
2		suppssfv.v	. . . . . . . . 9 |- ( ( ph /\ x e. D ) -> A e. V )
3		elex 2763	. . . . . . . . 9 |- ( A e. V -> A e. _V )
4	2, 3	syl 14	. . . . . . . 8 |- ( ( ph /\ x e. D ) -> A e. _V )
5	4	adantr 276	. . . . . . 7 |- ( ( ( ph /\ x e. D ) /\ ( F ` A ) =/= Z ) -> A e. _V )
6		suppssfv.f	. . . . . . . . . . 11 |- ( ph -> ( F ` Y ) = Z )
7		fveq2 5534	. . . . . . . . . . . 12 |- ( A = Y -> ( F ` A ) = ( F ` Y ) )
8	7	eqeq1d 2198	. . . . . . . . . . 11 |- ( A = Y -> ( ( F ` A ) = Z <-> ( F ` Y ) = Z ) )
9	6, 8	syl5ibrcom 157	. . . . . . . . . 10 |- ( ph -> ( A = Y -> ( F ` A ) = Z ) )
10	9	necon3d 2404	. . . . . . . . 9 |- ( ph -> ( ( F ` A ) =/= Z -> A =/= Y ) )
11	10	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. D ) -> ( ( F ` A ) =/= Z -> A =/= Y ) )
12	11	imp 124	. . . . . . 7 |- ( ( ( ph /\ x e. D ) /\ ( F ` A ) =/= Z ) -> A =/= Y )
13		eldifsn 3734	. . . . . . 7 |- ( A e. ( _V \ { Y } ) <-> ( A e. _V /\ A =/= Y ) )
14	5, 12, 13	sylanbrc 417	. . . . . 6 |- ( ( ( ph /\ x e. D ) /\ ( F ` A ) =/= Z ) -> A e. ( _V \ { Y } ) )
15	14	ex 115	. . . . 5 |- ( ( ph /\ x e. D ) -> ( ( F ` A ) =/= Z -> A e. ( _V \ { Y } ) ) )
16	1, 15	syl5 32	. . . 4 |- ( ( ph /\ x e. D ) -> ( ( F ` A ) e. ( _V \ { Z } ) -> A e. ( _V \ { Y } ) ) )
17	16	ss2rabdv 3251	. . 3 |- ( ph -> { x e. D | ( F ` A ) e. ( _V \ { Z } ) } C_ { x e. D | A e. ( _V \ { Y } ) } )
18		eqid 2189	. . . 4 |- ( x e. D |-> ( F ` A ) ) = ( x e. D |-> ( F ` A ) )
19	18	mptpreima 5140	. . 3 |- ( `' ( x e. D |-> ( F ` A ) ) " ( _V \ { Z } ) ) = { x e. D | ( F ` A ) e. ( _V \ { Z } ) }
20		eqid 2189	. . . 4 |- ( x e. D |-> A ) = ( x e. D |-> A )
21	20	mptpreima 5140	. . 3 |- ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) = { x e. D | A e. ( _V \ { Y } ) }
22	17, 19, 21	3sstr4g 3213	. 2 |- ( ph -> ( `' ( x e. D |-> ( F ` A ) ) " ( _V \ { Z } ) ) C_ ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) )
23		suppssfv.a	. 2 |- ( ph -> ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) C_ L )
24	22, 23	sstrd 3180	1 |- ( ph -> ( `' ( x e. D |-> ( F ` A ) ) " ( _V \ { Z } ) ) C_ L )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   =/= wne 2360   {crab 2472   _Vcvv 2752   \ cdif 3141   C_ wss 3144   {csn 3607   |-> cmpt 4079   `'ccnv 4643   "cima 4647   ` cfv 5235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fv 5243

This theorem is referenced by: (None)