Theorem zfrep6 6763

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 4574 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 4564. (Contributed by NM, 10-Oct-2003.)

Assertion

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

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

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem zfrep6

Step	Hyp	Ref	Expression
1		euex 2303	. . . . . . 7 |- ( E! y ph -> E. y ph )
2	1	ralimi 2860	. . . . . 6 |- ( A. x e. z E! y ph -> A. x e. z E. y ph )
3		rabid2 3044	. . . . . 6 |- ( z = { x e. z | E. y ph } <-> A. x e. z E. y ph )
4	2, 3	sylibr 212	. . . . 5 |- ( A. x e. z E! y ph -> z = { x e. z | E. y ph } )
5		19.42v 1949	. . . . . . 7 |- ( E. y ( x e. z /\ ph ) <-> ( x e. z /\ E. y ph ) )
6	5	abbii 2601	. . . . . 6 |- { x | E. y ( x e. z /\ ph ) } = { x | ( x e. z /\ E. y ph ) }
7		dmopab 5219	. . . . . 6 |- dom { <. x , y >. | ( x e. z /\ ph ) } = { x | E. y ( x e. z /\ ph ) }
8		df-rab 2826	. . . . . 6 |- { x e. z | E. y ph } = { x | ( x e. z /\ E. y ph ) }
9	6, 7, 8	3eqtr4i 2506	. . . . 5 |- dom { <. x , y >. | ( x e. z /\ ph ) } = { x e. z | E. y ph }
10	4, 9	syl6reqr 2527	. . . 4 |- ( A. x e. z E! y ph -> dom { <. x , y >. | ( x e. z /\ ph ) } = z )
11		vex 3121	. . . 4 |- z e. _V
12	10, 11	syl6eqel 2563	. . 3 |- ( A. x e. z E! y ph -> dom { <. x , y >. | ( x e. z /\ ph ) } e. _V )
13		eumo 2308	. . . . . . 7 |- ( E! y ph -> E* y ph )
14	13	imim2i 14	. . . . . 6 |- ( ( x e. z -> E! y ph ) -> ( x e. z -> E* y ph ) )
15		moanimv 2356	. . . . . 6 |- ( E* y ( x e. z /\ ph ) <-> ( x e. z -> E* y ph ) )
16	14, 15	sylibr 212	. . . . 5 |- ( ( x e. z -> E! y ph ) -> E* y ( x e. z /\ ph ) )
17	16	alimi 1614	. . . 4 |- ( A. x ( x e. z -> E! y ph ) -> A. x E* y ( x e. z /\ ph ) )
18		df-ral 2822	. . . 4 |- ( A. x e. z E! y ph <-> A. x ( x e. z -> E! y ph ) )
19		funopab 5627	. . . 4 |- ( Fun { <. x , y >. | ( x e. z /\ ph ) } <-> A. x E* y ( x e. z /\ ph ) )
20	17, 18, 19	3imtr4i 266	. . 3 |- ( A. x e. z E! y ph -> Fun { <. x , y >. | ( x e. z /\ ph ) } )
21		funrnex 6762	. . 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 60	. 2 |- ( A. x e. z E! y ph -> ran { <. x , y >. | ( x e. z /\ ph ) } e. _V )
23		nfra1 2848	. . 3 |- F/ x A. x e. z E! y ph
24	10	eleq2d 2537	. . . 4 |- ( A. x e. z E! y ph -> ( x e. dom { <. x , y >. | ( x e. z /\ ph ) } <-> x e. z ) )
25		opabid 4760	. . . . . . . . 9 |- ( <. x , y >. e. { <. x , y >. | ( x e. z /\ ph ) } <-> ( x e. z /\ ph ) )
26		vex 3121	. . . . . . . . . 10 |- x e. _V
27		vex 3121	. . . . . . . . . 10 |- y e. _V
28	26, 27	opelrn 5240	. . . . . . . . 9 |- ( <. x , y >. e. { <. x , y >. | ( x e. z /\ ph ) } -> y e. ran { <. x , y >. | ( x e. z /\ ph ) } )
29	25, 28	sylbir 213	. . . . . . . 8 |- ( ( x e. z /\ ph ) -> y e. ran { <. x , y >. | ( x e. z /\ ph ) } )
30	29	ex 434	. . . . . . 7 |- ( x e. z -> ( ph -> y e. ran { <. x , y >. | ( x e. z /\ ph ) } ) )
31	30	impac 621	. . . . . 6 |- ( ( x e. z /\ ph ) -> ( y e. ran { <. x , y >. | ( x e. z /\ ph ) } /\ ph ) )
32	31	eximi 1635	. . . . 5 |- ( E. y ( x e. z /\ ph ) -> E. y ( y e. ran { <. x , y >. | ( x e. z /\ ph ) } /\ ph ) )
33	7	abeq2i 2594	. . . . 5 |- ( x e. dom { <. x , y >. | ( x e. z /\ ph ) } <-> E. y ( x e. z /\ ph ) )
34		df-rex 2823	. . . . 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 266	. . . 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 229	. . 3 |- ( A. x e. z E! y ph -> ( x e. z -> E. y e. ran { <. x , y >. | ( x e. z /\ ph ) } ph ) )
37	23, 36	ralrimi 2867	. 2 |- ( A. x e. z E! y ph -> A. x e. z E. y e. ran { <. x , y >. | ( x e. z /\ ph ) } ph )
38		nfopab1 4519	. . . . . 6 |- F/_ x { <. x , y >. | ( x e. z /\ ph ) }
39	38	nfrn 5251	. . . . 5 |- F/_ x ran { <. x , y >. | ( x e. z /\ ph ) }
40	39	nfeq2 2646	. . . 4 |- F/ x w = ran { <. x , y >. | ( x e. z /\ ph ) }
41		nfcv 2629	. . . . 5 |- F/_ y w
42		nfopab2 4520	. . . . . 6 |- F/_ y { <. x , y >. | ( x e. z /\ ph ) }
43	42	nfrn 5251	. . . . 5 |- F/_ y ran { <. x , y >. | ( x e. z /\ ph ) }
44	41, 43	rexeqf 3060	. . . 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 ) )
45	40, 44	ralbid 2901	. . 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 ) )
46	45	spcegv 3204	. 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. w A. x e. z E. y e. w ph ) )
47	22, 37, 46	sylc 60	1 |- ( A. x e. z E! y ph -> E. w A. x e. z E. y e. w ph )

Syntax hints:   -> wi 4   /\ wa 369   A.wal 1377   = wceq 1379   E.wex 1596   e. wcel 1767   E!weu 2275   E*wmo 2276   {cab 2452   A.wral 2817   E.wrex 2818   {crab 2821   _Vcvv 3118   <.cop 4039   {copab 4510   dom cdm 5005   ran crn 5006   Fun wfun 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602

This theorem is referenced by:  bnj865  33461