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Theorem inf3lemd 8040
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8048 for detailed description. (Contributed by NM, 28-Oct-1996.)
Hypotheses
Ref Expression
inf3lem.1  |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
inf3lem.2  |-  F  =  ( rec ( G ,  (/) )  |`  om )
inf3lem.3  |-  A  e. 
_V
inf3lem.4  |-  B  e. 
_V
Assertion
Ref Expression
inf3lemd  |-  ( A  e.  om  ->  ( F `  A )  C_  x )
Distinct variable group:    x, y, w
Allowed substitution hints:    A( x, y, w)    B( x, y, w)    F( x, y, w)    G( x, y, w)

Proof of Theorem inf3lemd
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5864 . . . . 5  |-  ( A  =  (/)  ->  ( F `
 A )  =  ( F `  (/) ) )
2 inf3lem.1 . . . . . 6  |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
3 inf3lem.2 . . . . . 6  |-  F  =  ( rec ( G ,  (/) )  |`  om )
4 inf3lem.3 . . . . . 6  |-  A  e. 
_V
5 inf3lem.4 . . . . . 6  |-  B  e. 
_V
62, 3, 4, 5inf3lemb 8038 . . . . 5  |-  ( F `
 (/) )  =  (/)
71, 6syl6eq 2524 . . . 4  |-  ( A  =  (/)  ->  ( F `
 A )  =  (/) )
8 0ss 3814 . . . 4  |-  (/)  C_  x
97, 8syl6eqss 3554 . . 3  |-  ( A  =  (/)  ->  ( F `
 A )  C_  x )
109a1d 25 . 2  |-  ( A  =  (/)  ->  ( A  e.  om  ->  ( F `  A )  C_  x ) )
11 nnsuc 6695 . . . 4  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  E. v  e.  om  A  =  suc  v )
12 vex 3116 . . . . . . . . . 10  |-  v  e. 
_V
132, 3, 12, 5inf3lemc 8039 . . . . . . . . 9  |-  ( v  e.  om  ->  ( F `  suc  v )  =  ( G `  ( F `  v ) ) )
1413eleq2d 2537 . . . . . . . 8  |-  ( v  e.  om  ->  (
u  e.  ( F `
 suc  v )  <->  u  e.  ( G `  ( F `  v ) ) ) )
15 vex 3116 . . . . . . . . . 10  |-  u  e. 
_V
16 fvex 5874 . . . . . . . . . 10  |-  ( F `
 v )  e. 
_V
172, 3, 15, 16inf3lema 8037 . . . . . . . . 9  |-  ( u  e.  ( G `  ( F `  v ) )  <->  ( u  e.  x  /\  ( u  i^i  x )  C_  ( F `  v ) ) )
1817simplbi 460 . . . . . . . 8  |-  ( u  e.  ( G `  ( F `  v ) )  ->  u  e.  x )
1914, 18syl6bi 228 . . . . . . 7  |-  ( v  e.  om  ->  (
u  e.  ( F `
 suc  v )  ->  u  e.  x ) )
2019ssrdv 3510 . . . . . 6  |-  ( v  e.  om  ->  ( F `  suc  v ) 
C_  x )
21 fveq2 5864 . . . . . . 7  |-  ( A  =  suc  v  -> 
( F `  A
)  =  ( F `
 suc  v )
)
2221sseq1d 3531 . . . . . 6  |-  ( A  =  suc  v  -> 
( ( F `  A )  C_  x  <->  ( F `  suc  v
)  C_  x )
)
2320, 22syl5ibrcom 222 . . . . 5  |-  ( v  e.  om  ->  ( A  =  suc  v  -> 
( F `  A
)  C_  x )
)
2423rexlimiv 2949 . . . 4  |-  ( E. v  e.  om  A  =  suc  v  ->  ( F `  A )  C_  x )
2511, 24syl 16 . . 3  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  ( F `  A )  C_  x )
2625expcom 435 . 2  |-  ( A  =/=  (/)  ->  ( A  e.  om  ->  ( F `  A )  C_  x
) )
2710, 26pm2.61ine 2780 1  |-  ( A  e.  om  ->  ( F `  A )  C_  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   {crab 2818   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785    |-> cmpt 4505   suc csuc 4880    |` cres 5001   ` cfv 5586   omcom 6678   reccrdg 7072
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-om 6679  df-recs 7039  df-rdg 7073
This theorem is referenced by:  inf3lem2  8042  inf3lem3  8043  inf3lem6  8046
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