Theorem suppssfv 6177

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 3773	. . . . 5 |- ( ( F ` A ) e. ( _V \ { Z } ) -> ( F ` A ) =/= Z )
2		suppssfv.v	. . . . . . . . 9 |- ( ( ph /\ x e. D ) -> A e. V )
3		elex 2788	. . . . . . . . 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 5599	. . . . . . . . . . . 12 |- ( A = Y -> ( F ` A ) = ( F ` Y ) )
8	7	eqeq1d 2216	. . . . . . . . . . 11 |- ( A = Y -> ( ( F ` A ) = Z <-> ( F ` Y ) = Z ) )
9	6, 8	syl5ibrcom 157	. . . . . . . . . 10 |- ( ph -> ( A = Y -> ( F ` A ) = Z ) )
10	9	necon3d 2422	. . . . . . . . 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 3771	. . . . . . 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 3282	. . 3 |- ( ph -> { x e. D | ( F ` A ) e. ( _V \ { Z } ) } C_ { x e. D | A e. ( _V \ { Y } ) } )
18		eqid 2207	. . . 4 |- ( x e. D |-> ( F ` A ) ) = ( x e. D |-> ( F ` A ) )
19	18	mptpreima 5195	. . 3 |- ( `' ( x e. D |-> ( F ` A ) ) " ( _V \ { Z } ) ) = { x e. D | ( F ` A ) e. ( _V \ { Z } ) }
20		eqid 2207	. . . 4 |- ( x e. D |-> A ) = ( x e. D |-> A )
21	20	mptpreima 5195	. . 3 |- ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) = { x e. D | A e. ( _V \ { Y } ) }
22	17, 19, 21	3sstr4g 3244	. 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 3211	1 |- ( ph -> ( `' ( x e. D |-> ( F ` A ) ) " ( _V \ { Z } ) ) C_ L )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   =/= wne 2378   {crab 2490   _Vcvv 2776   \ cdif 3171   C_ wss 3174   {csn 3643   |-> cmpt 4121   `'ccnv 4692   "cima 4696   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-xp 4699  df-rel 4700  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fv 5298

This theorem is referenced by: (None)