Theorem scottexs 4864

Description: Theorem scheme version of scottex 4862. The collection of all x of minimum rank such that ph(x) is true, is a set.

Assertion

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

Distinct variable groups:   x,y   ph,y

Proof of Theorem scottexs

Step	Hyp	Ref	Expression
1		ax-17 1007	. . . 4 |- (y e. {x | ph} -> A.z y e. {x | ph})
2		hbab1 1508	. . . 4 |- (y e. {x | ph} -> A.x y e. {x | ph})
3		ax-17 1007	. . . . 5 |- ((rank` z) (_ (rank` y) -> A.x(rank` z) (_ (rank` y))
4	2, 3	hbral 1732	. . . 4 |- (A.y e. {x | ph} (rank` z) (_ (rank` y) -> A.xA.y e. {x | ph} (rank` z) (_ (rank` y))
5		ax-17 1007	. . . 4 |- (A.y e. {x | ph} (rank` x) (_ (rank` y) -> A.zA.y e. {x | ph} (rank` x) (_ (rank` y))
6		fveq2 3835	. . . . . 6 |- (z = x -> (rank` z) = (rank` x))
7	6	sseq1d 2140	. . . . 5 |- (z = x -> ((rank` z) (_ (rank` y) <-> (rank` x) (_ (rank` y)))
8	7	ralbidv 1709	. . . 4 |- (z = x -> (A.y e. {x | ph} (rank` z) (_ (rank` y) <-> A.y e. {x | ph} (rank` x) (_ (rank` y)))
9	1, 2, 4, 5, 8	cbvrab 1956	. . 3 |- {z e. {x | ph} | A.y e. {x | ph} (rank` z) (_ (rank` y)} = {x e. {x | ph} | A.y e. {x | ph} (rank` x) (_ (rank` y)}
10		df-rab 1698	. . 3 |- {x e. {x | ph} | A.y e. {x | ph} (rank` x) (_ (rank` y)} = {x | (x e. {x | ph} /\ A.y e. {x | ph} (rank` x) (_ (rank` y))}
11		abid 1507	. . . . 5 |- (x e. {x | ph} <-> ph)
12		df-ral 1695	. . . . . 6 |- (A.y e. {x | ph} (rank` x) (_ (rank` y) <-> A.y(y e. {x | ph} -> (rank` x) (_ (rank` y)))
13		df-clab 1506	. . . . . . . 8 |- (y e. {x | ph} <-> [y / x]ph)
14	13	imbi1i 184	. . . . . . 7 |- ((y e. {x | ph} -> (rank` x) (_ (rank` y)) <-> ([y / x]ph -> (rank` x) (_ (rank` y)))
15	14	albii 1035	. . . . . 6 |- (A.y(y e. {x | ph} -> (rank` x) (_ (rank` y)) <-> A.y([y / x]ph -> (rank` x) (_ (rank` y)))
16	12, 15	bitri 171	. . . . 5 |- (A.y e. {x | ph} (rank` x) (_ (rank` y) <-> A.y([y / x]ph -> (rank` x) (_ (rank` y)))
17	11, 16	anbi12i 485	. . . 4 |- ((x e. {x | ph} /\ A.y e. {x | ph} (rank` x) (_ (rank` y)) <-> (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y))))
18	17	abbii 1618	. . 3 |- {x | (x e. {x | ph} /\ A.y e. {x | ph} (rank` x) (_ (rank` y))} = {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))}
19	9, 10, 18	3eqtri 1542	. 2 |- {z e. {x | ph} | A.y e. {x | ph} (rank` z) (_ (rank` y)} = {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))}
20		scottex 4862	. 2 |- {z e. {x | ph} | A.y e. {x | ph} (rank` z) (_ (rank` y)} e. V
21	19, 20	eqeltrri 1588	1 |- {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))} e. V

Syntax hints:   -> wi 3   /\ wa 221  A.wal 990   = wceq 992   e. wcel 994  [wsbc 1207  {cab 1505  A.wral 1691  {crab 1694  Vcvv 1857   (_ wss 2099  ` cfv 3263  rankcrnk 4788

This theorem is referenced by:  hta 4874

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-reg 4736  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-fv 3279  df-rdg 4233  df-r1 4789  df-rank 4790

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