Theorem scottexs 4690

Description: Theorem scheme version of scottex 4688. 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-17 968	. . . 4 |- (y e. {x | ph} -> A.z y e. {x | ph})
2		hbab1 1459	. . . 4 |- (y e. {x | ph} -> A.x y e. {x | ph})
3		ax-17 968	. . . . 5 |- ((rank` z) (_ (rank` y) -> A.x(rank` z) (_ (rank` y))
4	2, 3	hbral 1678	. . . 4 |- (A.y e. {x | ph} (rank` z) (_ (rank` y) -> A.xA.y e. {x | ph} (rank` z) (_ (rank` y))
5		ax-17 968	. . . 4 |- (A.y e. {x | ph} (rank` x) (_ (rank` y) -> A.zA.y e. {x | ph} (rank` x) (_ (rank` y))
6		fveq2 3709	. . . . . 6 |- (z = x -> (rank` z) = (rank` x))
7	6	sseq1d 2078	. . . . 5 |- (z = x -> ((rank` z) (_ (rank` y) <-> (rank` x) (_ (rank` y)))
8	7	ralbidv 1655	. . . 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 1901	. . 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 1644	. . 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 1458	. . . . 5 |- (x e. {x | ph} <-> ph)
12		df-ral 1641	. . . . . 6 |- (A.y e. {x | ph} (rank` x) (_ (rank` y) <-> A.y(y e. {x | ph} -> (rank` x) (_ (rank` y)))
13		df-clab 1457	. . . . . . . 8 |- (y e. {x | ph} <-> [y / x]ph)
14	13	imbi1i 186	. . . . . . 7 |- ((y e. {x | ph} -> (rank` x) (_ (rank` y)) <-> ([y / x]ph -> (rank` x) (_ (rank` y)))
15	14	albii 996	. . . . . 6 |- (A.y(y e. {x | ph} -> (rank` x) (_ (rank` y)) <-> A.y([y / x]ph -> (rank` x) (_ (rank` y)))
16	12, 15	bitr 173	. . . . 5 |- (A.y e. {x | ph} (rank` x) (_ (rank` y) <-> A.y([y / x]ph -> (rank` x) (_ (rank` y)))
17	11, 16	anbi12i 481	. . . 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 1567	. . 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	3eqtr 1491	. 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 4688	. 2 |- {z e. {x | ph} | A.y e. {x | ph} (rank` z) (_ (rank` y)} e. V
21	19, 20	eqeltrr 1537	1 |- {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))} e. V

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  [wsbc 1166  {cab 1456  A.wral 1637  {crab 1640  Vcvv 1802   (_ wss 2037  ` cfv 3172  rankcrnk 4614

This theorem is referenced by:  hta 4700

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-r1 4615  df-rank 4616

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