Theorem scottexs 7775

Description: Theorem scheme version of scottex 7773. 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 2548	. . . 4 |- F/_ z { x | ph }
2		nfab1 2550	. . . 4 |- F/_ x { x | ph }
3		nfv 1626	. . . . 5 |- F/ x ( rank ` z ) C_ ( rank ` y )
4	2, 3	nfral 2727	. . . 4 |- F/ x A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y )
5		nfv 1626	. . . 4 |- F/ z A. y e. { x | ph } ( rank ` x ) C_ ( rank ` y )
6		fveq2 5695	. . . . . 6 |- ( z = x -> ( rank ` z ) = ( rank ` x ) )
7	6	sseq1d 3343	. . . . 5 |- ( z = x -> ( ( rank ` z ) C_ ( rank ` y ) <-> ( rank ` x ) C_ ( rank ` y ) ) )
8	7	ralbidv 2694	. . . 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 2922	. . 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 2683	. . 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 2400	. . . . 5 |- ( x e. { x | ph } <-> ph )
12		df-ral 2679	. . . . . 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 3130	. . . . . . . 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 1572	. . . . . 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 2524	. . 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 2436	. 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 7773	. 2 |- { z e. { x | ph } | A. y e. { x | ph } ( rank ` z ) C_ ( rank ` y ) } e. _V
21	19, 20	eqeltrri 2483	1 |- { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) } e. _V

Syntax hints:   -> wi 4   /\ wa 359   A.wal 1546   e. wcel 1721   {cab 2398   A.wral 2674   {crab 2678   _Vcvv 2924   [.wsbc 3129   C_ wss 3288   ` cfv 5421   rankcrnk 7653

This theorem is referenced by:  hta  7785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-reg 7524  ax-inf2 7560

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-recs 6600  df-rdg 6635  df-r1 7654  df-rank 7655