Theorem bnj82 13210

Description: Technical lemma of bnj75 13310. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem).

Hypothesis

Ref	Expression
bnj82.1	|- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))

Assertion

Ref	Expression
bnj82	|- ([z / n]ps <-> A.i e. om (suc i e. z -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))

Distinct variable groups:   A,n   R,n   f,n   i,n   z,i   y,n

Proof of Theorem bnj82

Step	Hyp	Ref	Expression
1		bnj82.1	. . 3 |- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
2	1	sbbii 1538	. 2 |- ([z / n]ps <-> [z / n]A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
3		visset 2295	. . 3 |- z e. _V
4		sbcralg 2531	. . 3 |- (z e. _V -> ([z / n]A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)) <-> A.i e. om [z / n](suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R))))
5	3, 4	ax-mp 7	. 2 |- ([z / n]A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)) <-> A.i e. om [z / n](suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
6		sbim 1604	. . . 4 |- ([z / n](suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)) <-> ([z / n]suc i e. n -> [z / n](f` suc i) = U_y e. (f` i) pred(y, A, R)))
7		sbcel2gv 2512	. . . . . 6 |- (z e. _V -> ([z / n]suc i e. n <-> suc i e. z))
8	3, 7	ax-mp 7	. . . . 5 |- ([z / n]suc i e. n <-> suc i e. z)
9		ax-17 1317	. . . . . 6 |- ((f` suc i) = U_y e. (f` i) pred(y, A, R) -> A.n(f` suc i) = U_y e. (f` i) pred(y, A, R))
10	9	sbf 1551	. . . . 5 |- ([z / n](f` suc i) = U_y e. (f` i) pred(y, A, R) <-> (f` suc i) = U_y e. (f` i) pred(y, A, R))
11	8, 10	imbi12i 205	. . . 4 |- (([z / n]suc i e. n -> [z / n](f` suc i) = U_y e. (f` i) pred(y, A, R)) <-> (suc i e. z -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
12	6, 11	bitri 190	. . 3 |- ([z / n](suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)) <-> (suc i e. z -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
13	12	ralbii 2127	. 2 |- (A.i e. om [z / n](suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)) <-> A.i e. om (suc i e. z -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
14	2, 5, 13	3bitri 194	1 |- ([z / n]ps <-> A.i e. om (suc i e. z -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  _Vcvv 2292  U_ciun 3255  suc csuc 3659  omcom 3949  ` cfv 3998   predsyn-bnj14 12023

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-sbc 2454

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