Theorem scottexs 7557

Description: Theorem scheme version of scottex 7555. The collection of all x of minimum rank such that ph ( x ) is true, is a set. (Contributed by NM, 13-Oct-2003.)

Assertion

Ref	Expression
scottexs	|- { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) } e. _V

Distinct variable groups:   x, y   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem scottexs

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2419	. . . 4 |- F/_ z { x | ph }
2		nfab1 2421	. . . 4 |- F/_ x { x | ph }
3		nfv 1605	. . . . 5 |- F/ x ( rank ` z ) C_ ( rank ` y )
4	2, 3	nfral 2596	. . . 4 |- F/ x A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y )
5		nfv 1605	. . . 4 |- F/ z A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y )
6		fveq2 5525	. . . . . 6 |- ( z = x -> ( rank ` z ) = ( rank ` x ) )
7	6	sseq1d 3205	. . . . 5 |- ( z = x -> ( ( rank ` z ) C_ ( rank ` y ) <-> ( rank ` x ) C_ ( rank ` y ) ) )
8	7	ralbidv 2563	. . . 4 |- ( z = x -> ( A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y ) <-> A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) ) )
9	1, 2, 4, 5, 8	cbvrab 2786	. . 3 |- { z e. { x | ph } | A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y ) } = { x e. { x | ph } | A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) }
10		df-rab 2552	. . 3 |- { x e. { x | ph } | A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) } = { x | ( x e. { x | ph } /\ A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) ) }
11		abid 2271	. . . . 5 |- ( x e. { x | ph } <-> ph )
12		df-ral 2548	. . . . . 6 |- ( A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) <-> A. y ( y e. { x | ph } -> ( rank ` x ) C_ ( rank ` y ) ) )
13		df-sbc 2992	. . . . . . . 8 |- ( [. y / x ]. ph <-> y e. { x | ph } )
14	13	imbi1i 315	. . . . . . 7 |- ( ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) <-> ( y e. { x | ph } -> ( rank ` x ) C_ ( rank ` y ) ) )
15	14	albii 1553	. . . . . 6 |- ( A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) <-> A. y ( y e. { x | ph } -> ( rank ` x ) C_ ( rank ` y ) ) )
16	12, 15	bitr4i 243	. . . . 5 |- ( A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) <-> A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) )
17	11, 16	anbi12i 678	. . . 4 |- ( ( x e. { x | ph } /\ A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) ) <-> ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) )
18	17	abbii 2395	. . 3 |- { x | ( x e. { x | ph } /\ A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y ) ) } = { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) }
19	9, 10, 18	3eqtri 2307	. 2 |- { z e. { x | ph } | A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y ) } = { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) }
20		scottex 7555	. 2 |- { z e. { x | ph } | A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y ) } e. _V
21	19, 20	eqeltrri 2354	1 |- { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) } e. _V

Syntax hints:   -> wi 4   /\ wa 358   A.wal 1527   e. wcel 1684   {cab 2269   A.wral 2543   {crab 2547   _Vcvv 2788   [.wsbc 2991   C_ wss 3152   ` cfv 5255   rankcrnk 7435

This theorem is referenced by:  hta  7567

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436  df-rank 7437