Theorem zfrep6 4545

Description: A version of the Axiom of Replacement. Normally ph would have free variables x and y. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 3438 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 3428.

Assertion

Ref	Expression
zfrep6	|- (A.x e. z E!yph -> E.wA.x e. z E.y e. w ph)

Distinct variable groups:   ph,w   x,y,z,w

Proof of Theorem zfrep6

Step	Hyp	Ref	Expression
1		euex 1788	. . . . . . 7 |- (E!yph -> E.yph)
2	1	ralimi 2168	. . . . . 6 |- (A.x e. z E!yph -> A.x e. z E.yph)
3		rabid2 2254	. . . . . 6 |- (z = {x e. z | E.yph} <-> A.x e. z E.yph)
4	2, 3	sylibr 217	. . . . 5 |- (A.x e. z E!yph -> z = {x e. z | E.yph})
5		19.42v 1688	. . . . . . 7 |- (E.y(x e. z /\ ph) <-> (x e. z /\ E.yph))
6	5	abbii 2006	. . . . . 6 |- {x | E.y(x e. z /\ ph)} = {x | (x e. z /\ E.yph)}
7		dmopab 4167	. . . . . 6 |- dom {<.x, y>. | (x e. z /\ ph)} = {x | E.y(x e. z /\ ph)}
8		df-rab 2112	. . . . . 6 |- {x e. z | E.yph} = {x | (x e. z /\ E.yph)}
9	6, 7, 8	3eqtr4i 1921	. . . . 5 |- dom {<.x, y>. | (x e. z /\ ph)} = {x e. z | E.yph}
10	4, 9	syl6reqr 1947	. . . 4 |- (A.x e. z E!yph -> dom {<.x, y>. | (x e. z /\ ph)} = z)
11		visset 2295	. . . 4 |- z e. _V
12	10, 11	syl6eqel 1979	. . 3 |- (A.x e. z E!yph -> dom {<.x, y>. | (x e. z /\ ph)} e. _V)
13		eumo 1807	. . . . . . 7 |- (E!yph -> E*yph)
14	13	imim2i 11	. . . . . 6 |- ((x e. z -> E!yph) -> (x e. z -> E*yph))
15		moanimv 1829	. . . . . 6 |- (E*y(x e. z /\ ph) <-> (x e. z -> E*yph))
16	14, 15	sylibr 217	. . . . 5 |- ((x e. z -> E!yph) -> E*y(x e. z /\ ph))
17	16	alimi 1338	. . . 4 |- (A.x(x e. z -> E!yph) -> A.xE*y(x e. z /\ ph))
18		df-ral 2109	. . . 4 |- (A.x e. z E!yph <-> A.x(x e. z -> E!yph))
19		funopab 4455	. . . 4 |- (Fun {<.x, y>. | (x e. z /\ ph)} <-> A.xE*y(x e. z /\ ph))
20	17, 18, 19	3imtr4i 236	. . 3 |- (A.x e. z E!yph -> Fun {<.x, y>. | (x e. z /\ ph)})
21		funrnex 4544	. . 3 |- (dom {<.x, y>. | (x e. z /\ ph)} e. _V -> (Fun {<.x, y>. | (x e. z /\ ph)} -> ran {<.x, y>. | (x e. z /\ ph)} e. _V))
22	12, 20, 21	sylc 83	. 2 |- (A.x e. z E!yph -> ran {<.x, y>. | (x e. z /\ ph)} e. _V)
23		hbra1 2147	. . 3 |- (A.x e. z E!yph -> A.xA.x e. z E!yph)
24	10	eleq2d 1964	. . . 4 |- (A.x e. z E!yph -> (x e. dom {<.x, y>. | (x e. z /\ ph)} <-> x e. z))
25		opabid 3557	. . . . . . . . 9 |- (<.x, y>. e. {<.x, y>. | (x e. z /\ ph)} <-> (x e. z /\ ph))
26		visset 2295	. . . . . . . . . 10 |- x e. _V
27		visset 2295	. . . . . . . . . 10 |- y e. _V
28	26, 27	opelrn 4192	. . . . . . . . 9 |- (<.x, y>. e. {<.x, y>. | (x e. z /\ ph)} -> y e. ran {<.x, y>. | (x e. z /\ ph)})
29	25, 28	sylbir 218	. . . . . . . 8 |- ((x e. z /\ ph) -> y e. ran {<.x, y>. | (x e. z /\ ph)})
30	29	ex 402	. . . . . . 7 |- (x e. z -> (ph -> y e. ran {<.x, y>. | (x e. z /\ ph)}))
31	30	impac 423	. . . . . 6 |- ((x e. z /\ ph) -> (y e. ran {<.x, y>. | (x e. z /\ ph)} /\ ph))
32	31	eximi 1387	. . . . 5 |- (E.y(x e. z /\ ph) -> E.y(y e. ran {<.x, y>. | (x e. z /\ ph)} /\ ph))
33	7	abeq2i 2001	. . . . 5 |- (x e. dom {<.x, y>. | (x e. z /\ ph)} <-> E.y(x e. z /\ ph))
34		df-rex 2110	. . . . 5 |- (E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph <-> E.y(y e. ran {<.x, y>. | (x e. z /\ ph)} /\ ph))
35	32, 33, 34	3imtr4i 236	. . . 4 |- (x e. dom {<.x, y>. | (x e. z /\ ph)} -> E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph)
36	24, 35	syl6bir 232	. . 3 |- (A.x e. z E!yph -> (x e. z -> E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph))
37	23, 36	r19.21ai 2174	. 2 |- (A.x e. z E!yph -> A.x e. z E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph)
38		ax-17 1317	. . 3 |- (v e. ran {<.x, y>. | (x e. z /\ ph)} -> A.w v e. ran {<.x, y>. | (x e. z /\ ph)})
39		ax-17 1317	. . 3 |- (A.x e. z E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph -> A.wA.x e. z E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph)
40		hbopab1 3562	. . . . . 6 |- (w e. {<.x, y>. | (x e. z /\ ph)} -> A.x w e. {<.x, y>. | (x e. z /\ ph)})
41	40	hbrn 4198	. . . . 5 |- (w e. ran {<.x, y>. | (x e. z /\ ph)} -> A.x w e. ran {<.x, y>. | (x e. z /\ ph)})
42	41	hbeleq 1997	. . . 4 |- (w = ran {<.x, y>. | (x e. z /\ ph)} -> A.x w = ran {<.x, y>. | (x e. z /\ ph)})
43		ax-17 1317	. . . . 5 |- (v e. w -> A.y v e. w)
44		hbopab2 3563	. . . . . 6 |- (v e. {<.x, y>. | (x e. z /\ ph)} -> A.y v e. {<.x, y>. | (x e. z /\ ph)})
45	44	hbrn 4198	. . . . 5 |- (v e. ran {<.x, y>. | (x e. z /\ ph)} -> A.y v e. ran {<.x, y>. | (x e. z /\ ph)})
46	43, 45	rexeqf 2264	. . . 4 |- (w = ran {<.x, y>. | (x e. z /\ ph)} -> (E.y e. w ph <-> E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph))
47	42, 46	ralbid 2121	. . 3 |- (w = ran {<.x, y>. | (x e. z /\ ph)} -> (A.x e. z E.y e. w ph <-> A.x e. z E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph))
48	38, 39, 47	cla4egf 2362	. 2 |- (ran {<.x, y>. | (x e. z /\ ph)} e. _V -> (A.x e. z E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph -> E.wA.x e. z E.y e. w ph))
49	22, 37, 48	sylc 83	1 |- (A.x e. z E!yph -> E.wA.x e. z E.y e. w ph)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  E*wmo 1772  {cab 1871  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292  <.cop 3046  {copab 3395  dom cdm 3986  ran crn 3987  Fun wfun 3992

This theorem is referenced by:  bnj1337 13064  bnj854 13314

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-13 1311  ax-14 1312  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  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009

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