Theorem zfrep6 6780

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 4518 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 4508. (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 2343	. . . . . . 7 |- ( E! y ph -> E. y ph )
2	1	ralimi 2796	. . . . . 6 |- ( A. x e. z E! y ph -> A. x e. z E. y ph )
3		rabid2 2954	. . . . . 6 |- ( z = { x e. z | E. y ph } <-> A. x e. z E. y ph )
4	2, 3	sylibr 217	. . . . 5 |- ( A. x e. z E! y ph -> z = { x e. z | E. y ph } )
5		19.42v 1842	. . . . . . 7 |- ( E. y ( x e. z /\ ph ) <-> ( x e. z /\ E. y ph ) )
6	5	abbii 2587	. . . . . 6 |- { x | E. y ( x e. z /\ ph ) } = { x | ( x e. z /\ E. y ph ) }
7		dmopab 5051	. . . . . 6 |- dom { <. x , y >. | ( x e. z /\ ph ) } = { x | E. y ( x e. z /\ ph ) }
8		df-rab 2765	. . . . . 6 |- { x e. z | E. y ph } = { x | ( x e. z /\ E. y ph ) }
9	6, 7, 8	3eqtr4i 2503	. . . . 5 |- dom { <. x , y >. | ( x e. z /\ ph ) } = { x e. z | E. y ph }
10	4, 9	syl6reqr 2524	. . . 4 |- ( A. x e. z E! y ph -> dom { <. x , y >. | ( x e. z /\ ph ) } = z )
11		vex 3034	. . . 4 |- z e. _V
12	10, 11	syl6eqel 2557	. . 3 |- ( A. x e. z E! y ph -> dom { <. x , y >. | ( x e. z /\ ph ) } e. _V )
13		eumo 2348	. . . . . . 7 |- ( E! y ph -> E* y ph )
14	13	imim2i 16	. . . . . 6 |- ( ( x e. z -> E! y ph ) -> ( x e. z -> E* y ph ) )
15		moanimv 2380	. . . . . 6 |- ( E* y ( x e. z /\ ph ) <-> ( x e. z -> E* y ph ) )
16	14, 15	sylibr 217	. . . . 5 |- ( ( x e. z -> E! y ph ) -> E* y ( x e. z /\ ph ) )
17	16	alimi 1692	. . . 4 |- ( A. x ( x e. z -> E! y ph ) -> A. x E* y ( x e. z /\ ph ) )
18		df-ral 2761	. . . 4 |- ( A. x e. z E! y ph <-> A. x ( x e. z -> E! y ph ) )
19		funopab 5622	. . . 4 |- ( Fun { <. x , y >. | ( x e. z /\ ph ) } <-> A. x E* y ( x e. z /\ ph ) )
20	17, 18, 19	3imtr4i 274	. . 3 |- ( A. x e. z E! y ph -> Fun { <. x , y >. | ( x e. z /\ ph ) } )
21		funrnex 6779	. . 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 61	. 2 |- ( A. x e. z E! y ph -> ran { <. x , y >. | ( x e. z /\ ph ) } e. _V )
23		nfra1 2785	. . 3 |- F/ x A. x e. z E! y ph
24	10	eleq2d 2534	. . . 4 |- ( A. x e. z E! y ph -> ( x e. dom { <. x , y >. | ( x e. z /\ ph ) } <-> x e. z ) )
25		opabid 4708	. . . . . . . . 9 |- ( <. x , y >. e. { <. x , y >. | ( x e. z /\ ph ) } <-> ( x e. z /\ ph ) )
26		vex 3034	. . . . . . . . . 10 |- x e. _V
27		vex 3034	. . . . . . . . . 10 |- y e. _V
28	26, 27	opelrn 5072	. . . . . . . . 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 441	. . . . . . 7 |- ( x e. z -> ( ph -> y e. ran { <. x , y >. | ( x e. z /\ ph ) } ) )
31	30	impac 633	. . . . . 6 |- ( ( x e. z /\ ph ) -> ( y e. ran { <. x , y >. | ( x e. z /\ ph ) } /\ ph ) )
32	31	eximi 1715	. . . . 5 |- ( E. y ( x e. z /\ ph ) -> E. y ( y e. ran { <. x , y >. | ( x e. z /\ ph ) } /\ ph ) )
33	7	abeq2i 2583	. . . . 5 |- ( x e. dom { <. x , y >. | ( x e. z /\ ph ) } <-> E. y ( x e. z /\ ph ) )
34		df-rex 2762	. . . . 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 274	. . . 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 237	. . 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 2800	. 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 4462	. . . . . 6 |- F/_ x { <. x , y >. | ( x e. z /\ ph ) }
39	38	nfrn 5083	. . . . 5 |- F/_ x ran { <. x , y >. | ( x e. z /\ ph ) }
40	39	nfeq2 2627	. . . 4 |- F/ x w = ran { <. x , y >. | ( x e. z /\ ph ) }
41		nfcv 2612	. . . . 5 |- F/_ y w
42		nfopab2 4463	. . . . . 6 |- F/_ y { <. x , y >. | ( x e. z /\ ph ) }
43	42	nfrn 5083	. . . . 5 |- F/_ y ran { <. x , y >. | ( x e. z /\ ph ) }
44	41, 43	rexeqf 2970	. . . 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 2826	. . 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 3121	. 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 61	1 |- ( A. x e. z E! y ph -> E. w A. x e. z E. y e. w ph )

Syntax hints:   -> wi 4   /\ wa 376   A.wal 1450   = wceq 1452   E.wex 1671   e. wcel 1904   E!weu 2319   E*wmo 2320   {cab 2457   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031   <.cop 3965   {copab 4453   dom cdm 4839   ran crn 4840   Fun wfun 5583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597

This theorem is referenced by:  bnj865  29806