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Theorem tfri1dALT 6365
Description: Alternate proof of tfri1d 6349 in terms of tfr1on 6364.

Although this does show that the tfr1on 6364 proof is general enough to also prove tfri1d 6349, the tfri1d 6349 proof is simpler in places because it does not need to deal with 
X being any ordinal. For that reason, we have both proofs. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Jim Kingdon, 20-Mar-2022.)

Hypotheses
Ref Expression
tfri1dALT.1  |-  F  = recs ( G )
tfri1dALT.2  |-  ( ph  ->  A. x ( Fun 
G  /\  ( G `  x )  e.  _V ) )
Assertion
Ref Expression
tfri1dALT  |-  ( ph  ->  F  Fn  On )
Distinct variable group:    x, G
Allowed substitution hints:    ph( x)    F( x)

Proof of Theorem tfri1dALT
Dummy variables  z  a  b  c  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrfun 6334 . . . 4  |-  Fun recs ( G )
2 tfri1dALT.1 . . . . 5  |-  F  = recs ( G )
32funeqi 5249 . . . 4  |-  ( Fun 
F  <->  Fun recs ( G ) )
41, 3mpbir 146 . . 3  |-  Fun  F
54a1i 9 . 2  |-  ( ph  ->  Fun  F )
6 eqid 2187 . . . . . 6  |-  { a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) }  =  { a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) }
76tfrlem8 6332 . . . . 5  |-  Ord  dom recs ( G )
82dmeqi 4840 . . . . . 6  |-  dom  F  =  dom recs ( G )
9 ordeq 4384 . . . . . 6  |-  ( dom 
F  =  dom recs ( G )  ->  ( Ord  dom  F  <->  Ord  dom recs ( G ) ) )
108, 9ax-mp 5 . . . . 5  |-  ( Ord 
dom  F  <->  Ord  dom recs ( G
) )
117, 10mpbir 146 . . . 4  |-  Ord  dom  F
12 ordsson 4503 . . . 4  |-  ( Ord 
dom  F  ->  dom  F  C_  On )
1311, 12mp1i 10 . . 3  |-  ( ph  ->  dom  F  C_  On )
14 tfri1dALT.2 . . . . . . . . . 10  |-  ( ph  ->  A. x ( Fun 
G  /\  ( G `  x )  e.  _V ) )
15 simpl 109 . . . . . . . . . . 11  |-  ( ( Fun  G  /\  ( G `  x )  e.  _V )  ->  Fun  G )
1615alimi 1465 . . . . . . . . . 10  |-  ( A. x ( Fun  G  /\  ( G `  x
)  e.  _V )  ->  A. x Fun  G
)
1714, 16syl 14 . . . . . . . . 9  |-  ( ph  ->  A. x Fun  G
)
181719.21bi 1568 . . . . . . . 8  |-  ( ph  ->  Fun  G )
1918adantr 276 . . . . . . 7  |-  ( (
ph  /\  z  e.  On )  ->  Fun  G
)
20 ordon 4497 . . . . . . . 8  |-  Ord  On
2120a1i 9 . . . . . . 7  |-  ( (
ph  /\  z  e.  On )  ->  Ord  On )
22 simpr 110 . . . . . . . . . . 11  |-  ( ( Fun  G  /\  ( G `  x )  e.  _V )  ->  ( G `  x )  e.  _V )
2322alimi 1465 . . . . . . . . . 10  |-  ( A. x ( Fun  G  /\  ( G `  x
)  e.  _V )  ->  A. x ( G `
 x )  e. 
_V )
24 fveq2 5527 . . . . . . . . . . . 12  |-  ( x  =  f  ->  ( G `  x )  =  ( G `  f ) )
2524eleq1d 2256 . . . . . . . . . . 11  |-  ( x  =  f  ->  (
( G `  x
)  e.  _V  <->  ( G `  f )  e.  _V ) )
2625spv 1870 . . . . . . . . . 10  |-  ( A. x ( G `  x )  e.  _V  ->  ( G `  f
)  e.  _V )
2714, 23, 263syl 17 . . . . . . . . 9  |-  ( ph  ->  ( G `  f
)  e.  _V )
2827adantr 276 . . . . . . . 8  |-  ( (
ph  /\  z  e.  On )  ->  ( G `
 f )  e. 
_V )
29283ad2ant1 1019 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  On )  /\  y  e.  On  /\  f  Fn  y )  ->  ( G `  f )  e.  _V )
30 onsuc 4512 . . . . . . . . 9  |-  ( y  e.  On  ->  suc  y  e.  On )
31 unon 4522 . . . . . . . . 9  |-  U. On  =  On
3230, 31eleq2s 2282 . . . . . . . 8  |-  ( y  e.  U. On  ->  suc  y  e.  On )
3332adantl 277 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  On )  /\  y  e.  U. On )  ->  suc  y  e.  On )
34 onsuc 4512 . . . . . . . 8  |-  ( z  e.  On  ->  suc  z  e.  On )
3534adantl 277 . . . . . . 7  |-  ( (
ph  /\  z  e.  On )  ->  suc  z  e.  On )
362, 19, 21, 29, 33, 35tfr1on 6364 . . . . . 6  |-  ( (
ph  /\  z  e.  On )  ->  suc  z  C_ 
dom  F )
37 vex 2752 . . . . . . 7  |-  z  e. 
_V
3837sucid 4429 . . . . . 6  |-  z  e. 
suc  z
39 ssel2 3162 . . . . . 6  |-  ( ( suc  z  C_  dom  F  /\  z  e.  suc  z )  ->  z  e.  dom  F )
4036, 38, 39sylancl 413 . . . . 5  |-  ( (
ph  /\  z  e.  On )  ->  z  e. 
dom  F )
4140ex 115 . . . 4  |-  ( ph  ->  ( z  e.  On  ->  z  e.  dom  F
) )
4241ssrdv 3173 . . 3  |-  ( ph  ->  On  C_  dom  F )
4313, 42eqssd 3184 . 2  |-  ( ph  ->  dom  F  =  On )
44 df-fn 5231 . 2  |-  ( F  Fn  On  <->  ( Fun  F  /\  dom  F  =  On ) )
455, 43, 44sylanbrc 417 1  |-  ( ph  ->  F  Fn  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1361    = wceq 1363    e. wcel 2158   {cab 2173   A.wral 2465   E.wrex 2466   _Vcvv 2749    C_ wss 3141   U.cuni 3821   Ord word 4374   Oncon0 4375   suc csuc 4377   dom cdm 4638    |` cres 4640   Fun wfun 5222    Fn wfn 5223   ` cfv 5228  recscrecs 6318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-recs 6319
This theorem is referenced by: (None)
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