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Theorem inf3lemd 8076
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8084 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 5848 . . . . 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 8074 . . . . 5  |-  ( F `
 (/) )  =  (/)
71, 6syl6eq 2459 . . . 4  |-  ( A  =  (/)  ->  ( F `
 A )  =  (/) )
8 0ss 3767 . . . 4  |-  (/)  C_  x
97, 8syl6eqss 3491 . . 3  |-  ( A  =  (/)  ->  ( F `
 A )  C_  x )
109a1d 25 . 2  |-  ( A  =  (/)  ->  ( A  e.  om  ->  ( F `  A )  C_  x ) )
11 nnsuc 6699 . . . 4  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  E. v  e.  om  A  =  suc  v )
12 vex 3061 . . . . . . . . . 10  |-  v  e. 
_V
132, 3, 12, 5inf3lemc 8075 . . . . . . . . 9  |-  ( v  e.  om  ->  ( F `  suc  v )  =  ( G `  ( F `  v ) ) )
1413eleq2d 2472 . . . . . . . 8  |-  ( v  e.  om  ->  (
u  e.  ( F `
 suc  v )  <->  u  e.  ( G `  ( F `  v ) ) ) )
15 vex 3061 . . . . . . . . . 10  |-  u  e. 
_V
16 fvex 5858 . . . . . . . . . 10  |-  ( F `
 v )  e. 
_V
172, 3, 15, 16inf3lema 8073 . . . . . . . . 9  |-  ( u  e.  ( G `  ( F `  v ) )  <->  ( u  e.  x  /\  ( u  i^i  x )  C_  ( F `  v ) ) )
1817simplbi 458 . . . . . . . 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 3447 . . . . . 6  |-  ( v  e.  om  ->  ( F `  suc  v ) 
C_  x )
21 fveq2 5848 . . . . . . 7  |-  ( A  =  suc  v  -> 
( F `  A
)  =  ( F `
 suc  v )
)
2221sseq1d 3468 . . . . . 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 2889 . . . 4  |-  ( E. v  e.  om  A  =  suc  v  ->  ( F `  A )  C_  x )
2511, 24syl 17 . . 3  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  ( F `  A )  C_  x )
2625expcom 433 . 2  |-  ( A  =/=  (/)  ->  ( A  e.  om  ->  ( F `  A )  C_  x
) )
2710, 26pm2.61ine 2716 1  |-  ( A  e.  om  ->  ( F `  A )  C_  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754   {crab 2757   _Vcvv 3058    i^i cin 3412    C_ wss 3413   (/)c0 3737    |-> cmpt 4452    |` cres 4824   suc csuc 5411   ` cfv 5568   omcom 6682   reccrdg 7111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-om 6683  df-wrecs 7012  df-recs 7074  df-rdg 7112
This theorem is referenced by:  inf3lem2  8078  inf3lem3  8079  inf3lem6  8082
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