Theorem isarep1 3577

Description: Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by ph(x, y) i.e. the class ({<.x, y>. | ph}"A). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation.

Assertion

Ref	Expression
isarep1	|- (b e. ({<.x, y>. | ph}"A) <-> E.x e. A [b / y]ph)

Distinct variable groups:   x,A   x,b,y

Proof of Theorem isarep1

Step	Hyp	Ref	Expression
1		visset 1813	. . 3 |- b e. V
2	1	elima 3408	. 2 |- (b e. ({<.x, y>. | ph}"A) <-> E.z e. A z{<.x, y>. | ph}b)
3		df-br 2620	. . . 4 |- (z{<.x, y>. | ph}b <-> <.z, b>. e. {<.x, y>. | ph})
4		opabsb 2815	. . . 4 |- (<.z, b>. e. {<.x, y>. | ph} <-> [b / y][z / x]ph)
5	3, 4	bitr 173	. . 3 |- (z{<.x, y>. | ph}b <-> [b / y][z / x]ph)
6	5	rexbii 1668	. 2 |- (E.z e. A z{<.x, y>. | ph}b <-> E.z e. A [b / y][z / x]ph)
7		hbs1 1332	. . . 4 |- ([z / x]ph -> A.x[z / x]ph)
8	7	hbsb 1333	. . 3 |- ([b / y][z / x]ph -> A.x[b / y][z / x]ph)
9		ax-17 971	. . 3 |- ([b / y]ph -> A.z[b / y]ph)
10		sbequ12r 1182	. . . . 5 |- (z = x -> ([z / x]ph <-> ph))
11	10	sbcbidv 1977	. . . 4 |- ((z = x /\ b e. V) -> ([b / y][z / x]ph <-> [b / y]ph))
12	1, 11	mpan2 696	. . 3 |- (z = x -> ([b / y][z / x]ph <-> [b / y]ph))
13	8, 9, 12	cbvrex 1799	. 2 |- (E.z e. A [b / y][z / x]ph <-> E.x e. A [b / y]ph)
14	2, 6, 13	3bitr 177	1 |- (b e. ({<.x, y>. | ph}"A) <-> E.x e. A [b / y]ph)

Syntax hints:   <-> wb 146   = wceq 956   e. wcel 958  [wsbc 1170  E.wrex 1646  Vcvv 1811  <.cop 2411   class class class wbr 2619  {copab 2666  "cima 3173

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191

Copyright terms: Public domain