Theorem scottexs 7811

Description: Theorem scheme version of scottex 7809. 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 2572	. . . 4 |- F/_ z { x | ph }
2		nfab1 2574	. . . 4 |- F/_ x { x | ph }
3		nfv 1629	. . . . 5 |- F/ x ( rank ` z ) C_ ( rank ` y )
4	2, 3	nfral 2759	. . . 4 |- F/ x A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y )
5		nfv 1629	. . . 4 |- F/ z A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y )
6		fveq2 5728	. . . . . 6 |- ( z = x -> ( rank ` z ) = ( rank ` x ) )
7	6	sseq1d 3375	. . . . 5 |- ( z = x -> ( ( rank ` z ) C_ ( rank ` y ) <-> ( rank ` x ) C_ ( rank ` y ) ) )
8	7	ralbidv 2725	. . . 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 2954	. . 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 2714	. . 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 2424	. . . . 5 |- ( x e. { x | ph } <-> ph )
12		df-ral 2710	. . . . . 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 3162	. . . . . . . 8 |- ( [. y / x ]. ph <-> y e. { x | ph } )
14	13	imbi1i 316	. . . . . . 7 |- ( ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) <-> ( y e. { x | ph } -> ( rank ` x ) C_ ( rank ` y ) ) )
15	14	albii 1575	. . . . . 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 244	. . . . 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 679	. . . 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 2548	. . 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 2460	. 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 7809	. 2 |- { z e. { x | ph } | A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y ) } e. _V
21	19, 20	eqeltrri 2507	1 |- { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) } e. _V

Syntax hints:   -> wi 4   /\ wa 359   A.wal 1549   e. wcel 1725   {cab 2422   A.wral 2705   {crab 2709   _Vcvv 2956   [.wsbc 3161   C_ wss 3320   ` cfv 5454   rankcrnk 7689

This theorem is referenced by:  hta  7821

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-reg 7560  ax-inf2 7596

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668  df-r1 7690  df-rank 7691