Theorem suppssfv 6061

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 3713	. . . . 5 |- ( ( F ` A ) e. ( _V \ { Z } ) -> ( F ` A ) =/= Z )
2		suppssfv.v	. . . . . . . . 9 |- ( ( ph /\ x e. D ) -> A e. V )
3		elex 2742	. . . . . . . . 9 |- ( A e. V -> A e. _V )
4	2, 3	syl 14	. . . . . . . 8 |- ( ( ph /\ x e. D ) -> A e. _V )
5	4	adantr 274	. . . . . . 7 |- ( ( ( ph /\ x e. D ) /\ ( F ` A ) =/= Z ) -> A e. _V )
6		suppssfv.f	. . . . . . . . . . 11 |- ( ph -> ( F ` Y ) = Z )
7		fveq2 5499	. . . . . . . . . . . 12 |- ( A = Y -> ( F ` A ) = ( F ` Y ) )
8	7	eqeq1d 2180	. . . . . . . . . . 11 |- ( A = Y -> ( ( F ` A ) = Z <-> ( F ` Y ) = Z ) )
9	6, 8	syl5ibrcom 156	. . . . . . . . . 10 |- ( ph -> ( A = Y -> ( F ` A ) = Z ) )
10	9	necon3d 2385	. . . . . . . . 9 |- ( ph -> ( ( F ` A ) =/= Z -> A =/= Y ) )
11	10	adantr 274	. . . . . . . 8 |- ( ( ph /\ x e. D ) -> ( ( F ` A ) =/= Z -> A =/= Y ) )
12	11	imp 123	. . . . . . 7 |- ( ( ( ph /\ x e. D ) /\ ( F ` A ) =/= Z ) -> A =/= Y )
13		eldifsn 3711	. . . . . . 7 |- ( A e. ( _V \ { Y } ) <-> ( A e. _V /\ A =/= Y ) )
14	5, 12, 13	sylanbrc 415	. . . . . 6 |- ( ( ( ph /\ x e. D ) /\ ( F ` A ) =/= Z ) -> A e. ( _V \ { Y } ) )
15	14	ex 114	. . . . 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 3229	. . 3 |- ( ph -> { x e. D | ( F ` A ) e. ( _V \ { Z } ) } C_ { x e. D | A e. ( _V \ { Y } ) } )
18		eqid 2171	. . . 4 |- ( x e. D |-> ( F ` A ) ) = ( x e. D |-> ( F ` A ) )
19	18	mptpreima 5106	. . 3 |- ( `' ( x e. D |-> ( F ` A ) ) " ( _V \ { Z } ) ) = { x e. D | ( F ` A ) e. ( _V \ { Z } ) }
20		eqid 2171	. . . 4 |- ( x e. D |-> A ) = ( x e. D |-> A )
21	20	mptpreima 5106	. . 3 |- ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) = { x e. D | A e. ( _V \ { Y } ) }
22	17, 19, 21	3sstr4g 3191	. 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 3158	1 |- ( ph -> ( `' ( x e. D |-> ( F ` A ) ) " ( _V \ { Z } ) ) C_ L )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1349   e. wcel 2142   =/= wne 2341   {crab 2453   _Vcvv 2731   \ cdif 3119   C_ wss 3122   {csn 3584   |-> cmpt 4051   `'ccnv 4611   "cima 4615   ` cfv 5200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-ral 2454  df-rex 2455  df-rab 2458  df-v 2733  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-br 3991  df-opab 4052  df-mpt 4053  df-xp 4618  df-rel 4619  df-cnv 4620  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fv 5208

This theorem is referenced by: (None)