Theorem zfrep6 3554

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 2671 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 2661.

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		ax-17 1190	. . 3 |- (v e. ran {<.x, y>. | (x e. z /\ ph)} -> A.w v e. ran {<.x, y>. | (x e. z /\ ph)})
2		ax-17 1190	. . 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)
3		hbopab1 2775	. . . . . 6 |- (w e. {<.x, y>. | (x e. z /\ ph)} -> A.x w e. {<.x, y>. | (x e. z /\ ph)})
4	3	hbrn 3307	. . . . 5 |- (w e. ran {<.x, y>. | (x e. z /\ ph)} -> A.x w e. ran {<.x, y>. | (x e. z /\ ph)})
5	4	hbeleq 1543	. . . 4 |- (w = ran {<.x, y>. | (x e. z /\ ph)} -> A.x w = ran {<.x, y>. | (x e. z /\ ph)})
6		ax-17 1190	. . . . 5 |- (v e. w -> A.y v e. w)
7		hbopab2 2776	. . . . . 6 |- (v e. {<.x, y>. | (x e. z /\ ph)} -> A.y v e. {<.x, y>. | (x e. z /\ ph)})
8	7	hbrn 3307	. . . . 5 |- (v e. ran {<.x, y>. | (x e. z /\ ph)} -> A.y v e. ran {<.x, y>. | (x e. z /\ ph)})
9	6, 8	rexeq1f 1760	. . . 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))
10	5, 9	ralbid 1637	. . 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))
11	1, 2, 10	cla4egf 1836	. 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))
12		funrnex 3553	. . 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))
13		euex 1371	. . . . . . 7 |- (E!yph -> E.yph)
14	13	r19.20si 1682	. . . . . 6 |- (A.x e. z E!yph -> A.x e. z E.yph)
15		rabid2 1746	. . . . . 6 |- (z = {x e. z | E.yph} <-> A.x e. z E.yph)
16	14, 15	sylibr 200	. . . . 5 |- (A.x e. z E!yph -> z = {x e. z | E.yph})
17		19.42v 1290	. . . . . . 7 |- (E.y(x e. z /\ ph) <-> (x e. z /\ E.yph))
18	17	abbii 1551	. . . . . 6 |- {x | E.y(x e. z /\ ph)} = {x | (x e. z /\ E.yph)}
19		dmopab 3277	. . . . . 6 |- dom {<.x, y>. | (x e. z /\ ph)} = {x | E.y(x e. z /\ ph)}
20		df-rab 1628	. . . . . 6 |- {x e. z | E.yph} = {x | (x e. z /\ E.yph)}
21	18, 19, 20	3eqtr4 1481	. . . . 5 |- dom {<.x, y>. | (x e. z /\ ph)} = {x e. z | E.yph}
22	16, 21	syl6reqr 1502	. . . 4 |- (A.x e. z E!yph -> dom {<.x, y>. | (x e. z /\ ph)} = z)
23		visset 1788	. . . 4 |- z e. V
24	22, 23	syl6eqel 1532	. . 3 |- (A.x e. z E!yph -> dom {<.x, y>. | (x e. z /\ ph)} e. V)
25		eumo 1388	. . . . . . 7 |- (E!yph -> E*yph)
26	25	imim2i 17	. . . . . 6 |- ((x e. z -> E!yph) -> (x e. z -> E*yph))
27		moanimv 1406	. . . . . 6 |- (E*y(x e. z /\ ph) <-> (x e. z -> E*yph))
28	26, 27	sylibr 200	. . . . 5 |- ((x e. z -> E!yph) -> E*y(x e. z /\ ph))
29	28	19.20i 968	. . . 4 |- (A.x(x e. z -> E!yph) -> A.xE*y(x e. z /\ ph))
30		df-ral 1625	. . . 4 |- (A.x e. z E!yph <-> A.x(x e. z -> E!yph))
31		funopab 3488	. . . 4 |- (Fun {<.x, y>. | (x e. z /\ ph)} <-> A.xE*y(x e. z /\ ph))
32	29, 30, 31	3imtr4 219	. . 3 |- (A.x e. z E!yph -> Fun {<.x, y>. | (x e. z /\ ph)})
33	12, 24, 32	sylc 68	. 2 |- (A.x e. z E!yph -> ran {<.x, y>. | (x e. z /\ ph)} e. V)
34		hbra1 1663	. . 3 |- (A.x e. z E!yph -> A.xA.x e. z E!yph)
35	22	eleq2d 1517	. . . 4 |- (A.x e. z E!yph -> (x e. dom {<.x, y>. | (x e. z /\ ph)} <-> x e. z))
36		opabid 2772	. . . . . . . . 9 |- (<.x, y>. e. {<.x, y>. | (x e. z /\ ph)} <-> (x e. z /\ ph))
37		visset 1788	. . . . . . . . . 10 |- x e. V
38		visset 1788	. . . . . . . . . 10 |- y e. V
39	37, 38	opelrn 3303	. . . . . . . . 9 |- (<.x, y>. e. {<.x, y>. | (x e. z /\ ph)} -> y e. ran {<.x, y>. | (x e. z /\ ph)})
40	36, 39	sylbir 201	. . . . . . . 8 |- ((x e. z /\ ph) -> y e. ran {<.x, y>. | (x e. z /\ ph)})
41	40	ex 373	. . . . . . 7 |- (x e. z -> (ph -> y e. ran {<.x, y>. | (x e. z /\ ph)}))
42	41	impac 387	. . . . . 6 |- ((x e. z /\ ph) -> (y e. ran {<.x, y>. | (x e. z /\ ph)} /\ ph))
43	42	19.22i 1016	. . . . 5 |- (E.y(x e. z /\ ph) -> E.y(y e. ran {<.x, y>. | (x e. z /\ ph)} /\ ph))
44	19	abeq2i 1546	. . . . 5 |- (x e. dom {<.x, y>. | (x e. z /\ ph)} <-> E.y(x e. z /\ ph))
45		df-rex 1626	. . . . 5 |- (E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph <-> E.y(y e. ran {<.x, y>. | (x e. z /\ ph)} /\ ph))
46	43, 44, 45	3imtr4 219	. . . 4 |- (x e. dom {<.x, y>. | (x e. z /\ ph)} -> E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph)
47	35, 46	syl6bir 215	. . 3 |- (A.x e. z E!yph -> (x e. z -> E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph))
48	34, 47	r19.21ai 1688	. 2 |- (A.x e. z E!yph -> A.x e. z E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph)
49	11, 33, 48	sylc 68	1 |- (A.x e. z E!yph -> E.wA.x e. z E.y e. w ph)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105  E!weu 1357  E*wmo 1358  {cab 1440  A.wral 1621  E.wrex 1622  {crab 1624  Vcvv 1786  <.cop 2382  {copab 2634  dom cdm 3133  ran crn 3134  Fun wfun 3139

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-rab 1628  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-id 2797  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156

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