Theorem zfrep6 2744

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 zfaus 1480 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 1075.

Assertion

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

Distinct variable group(s):   ph,z,w   x,y,z,w

Proof of Theorem zfrep6

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 |- (v e. ran {<.x, y>. | (x e. z /\ ph)} -> A.w v e. ran {<.x, y>. | (x e. z /\ ph)})
2		ax-17 925	. . 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 2112	. . . . . 6 |- (w e. {<.x, y>. | (x e. z /\ ph)} -> A.x w e. {<.x, y>. | (x e. z /\ ph)})
4	3	hbrn 2564	. . . . 5 |- (w e. ran {<.x, y>. | (x e. z /\ ph)} -> A.x w e. ran {<.x, y>. | (x e. z /\ ph)})
5	4	hbeleq 1173	. . . 4 |- (w = ran {<.x, y>. | (x e. z /\ ph)} -> A.x w = ran {<.x, y>. | (x e. z /\ ph)})
6		ax-17 925	. . . . 5 |- (v e. w -> A.y v e. w)
7		hbopab2 2113	. . . . . 6 |- (v e. {<.x, y>. | (x e. z /\ ph)} -> A.y v e. {<.x, y>. | (x e. z /\ ph)})
8	7	hbrn 2564	. . . . 5 |- (v e. ran {<.x, y>. | (x e. z /\ ph)} -> A.y v e. ran {<.x, y>. | (x e. z /\ ph)})
9	6, 8	rexeqf 1322	. . . 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	birald 1217	. . 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 1395	. 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 2743	. . 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		visset 1350	. . . 4 |- z e. V
14		euex 1021	. . . . . . . 8 |- (E!yph -> E.yph)
15	14	r19.20si 1254	. . . . . . 7 |- (A.x e. z E!yph -> A.x e. z E.yph)
16		rabid2 1309	. . . . . . 7 |- (z = {x e. z | E.yph} <-> A.x e. z E.yph)
17	15, 16	sylibr 175	. . . . . 6 |- (A.x e. z E!yph -> z = {x e. z | E.yph})
18		19.42v 966	. . . . . . . 8 |- (E.y(x e. z /\ ph) <-> (x e. z /\ E.yph))
19	18	biabi 1181	. . . . . . 7 |- {x | E.y(x e. z /\ ph)} = {x | (x e. z /\ E.yph)}
20		dmopab 2539	. . . . . . 7 |- dom {<.x, y>. | (x e. z /\ ph)} = {x | E.y(x e. z /\ ph)}
21		df-rab 1208	. . . . . . 7 |- {x e. z | E.yph} = {x | (x e. z /\ E.yph)}
22	19, 20, 21	3eqtr4 1126	. . . . . 6 |- dom {<.x, y>. | (x e. z /\ ph)} = {x e. z | E.yph}
23	17, 22	syl6reqr 1143	. . . . 5 |- (A.x e. z E!yph -> dom {<.x, y>. | (x e. z /\ ph)} = z)
24	23	eleq1d 1155	. . . 4 |- (A.x e. z E!yph -> (dom {<.x, y>. | (x e. z /\ ph)} e. V <-> z e. V))
25	13, 24	mpbiri 169	. . 3 |- (A.x e. z E!yph -> dom {<.x, y>. | (x e. z /\ ph)} e. V)
26		eumo 1037	. . . . . . 7 |- (E!yph -> E*yph)
27	26	syl3 18	. . . . . 6 |- ((x e. z -> E!yph) -> (x e. z -> E*yph))
28		moanimv 1052	. . . . . 6 |- (E*y(x e. z /\ ph) <-> (x e. z -> E*yph))
29	27, 28	sylibr 175	. . . . 5 |- ((x e. z -> E!yph) -> E*y(x e. z /\ ph))
30	29	19.20i 691	. . . 4 |- (A.x(x e. z -> E!yph) -> A.xE*y(x e. z /\ ph))
31		df-ral 1205	. . . 4 |- (A.x e. z E!yph <-> A.x(x e. z -> E!yph))
32		funopab 2694	. . . 4 |- (Fun {<.x, y>. | (x e. z /\ ph)} <-> A.xE*y(x e. z /\ ph))
33	30, 31, 32	3imtr4 192	. . 3 |- (A.x e. z E!yph -> Fun {<.x, y>. | (x e. z /\ ph)})
34	12, 25, 33	sylc 62	. 2 |- (A.x e. z E!yph -> ran {<.x, y>. | (x e. z /\ ph)} e. V)
35		hbra1 1237	. . 3 |- (A.x e. z E!yph -> A.xA.x e. z E!yph)
36	23	eleq2d 1156	. . . 4 |- (A.x e. z E!yph -> (x e. dom {<.x, y>. | (x e. z /\ ph)} <-> x e. z))
37		opabid 2099	. . . . . . . . 9 |- (<.x, y>. e. {<.x, y>. | (x e. z /\ ph)} <-> (x e. z /\ ph))
38		visset 1350	. . . . . . . . . 10 |- x e. V
39		visset 1350	. . . . . . . . . 10 |- y e. V
40	38, 39	opelrn 2560	. . . . . . . . 9 |- (<.x, y>. e. {<.x, y>. | (x e. z /\ ph)} -> y e. ran {<.x, y>. | (x e. z /\ ph)})
41	37, 40	sylbir 176	. . . . . . . 8 |- ((x e. z /\ ph) -> y e. ran {<.x, y>. | (x e. z /\ ph)})
42	41	exp 291	. . . . . . 7 |- (x e. z -> (ph -> y e. ran {<.x, y>. | (x e. z /\ ph)}))
43	42	impac 304	. . . . . 6 |- ((x e. z /\ ph) -> (y e. ran {<.x, y>. | (x e. z /\ ph)} /\ ph))
44	43	19.22i 723	. . . . 5 |- (E.y(x e. z /\ ph) -> E.y(y e. ran {<.x, y>. | (x e. z /\ ph)} /\ ph))
45	20	cleqabi 1176	. . . . 5 |- (x e. dom {<.x, y>. | (x e. z /\ ph)} <-> E.y(x e. z /\ ph))
46		df-rex 1206	. . . . 5 |- (E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph <-> E.y(y e. ran {<.x, y>. | (x e. z /\ ph)} /\ ph))
47	44, 45, 46	3imtr4 192	. . . 4 |- (x e. dom {<.x, y>. | (x e. z /\ ph)} -> E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph)
48	36, 47	syl6bir 188	. . 3 |- (A.x e. z E!yph -> (x e. z -> E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph))
49	35, 48	r19.21ai 1258	. 2 |- (A.x e. z E!yph -> A.x e. z E.y e. ran {<.x, y>. | (x e. z /\ ph)}ph)
50	11, 34, 49	sylc 62	1 |- (A.x e. z E!yph -> E.wA.x e. z E.y e. w ph)

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   e. wel 803  E!weu 1007  E*wmo 1008  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  {crab 1204  Vcvv 1348  <.cop 1810  {copab 2055  dom cdm 2410  ran crn 2411  Fun wfun 2416

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433

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