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Theorem tfri1dALT 6216
Description: Alternate proof of tfri1d 6200 in terms of tfr1on 6215.

Although this does show that the tfr1on 6215 proof is general enough to also prove tfri1d 6200, the tfri1d 6200 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 6185 . . . 4  |-  Fun recs ( G )
2 tfri1dALT.1 . . . . 5  |-  F  = recs ( G )
32funeqi 5114 . . . 4  |-  ( Fun 
F  <->  Fun recs ( G ) )
41, 3mpbir 145 . . 3  |-  Fun  F
54a1i 9 . 2  |-  ( ph  ->  Fun  F )
6 eqid 2117 . . . . . 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 6183 . . . . 5  |-  Ord  dom recs ( G )
82dmeqi 4710 . . . . . 6  |-  dom  F  =  dom recs ( G )
9 ordeq 4264 . . . . . 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 145 . . . 4  |-  Ord  dom  F
12 ordsson 4378 . . . 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 108 . . . . . . . . . . 11  |-  ( ( Fun  G  /\  ( G `  x )  e.  _V )  ->  Fun  G )
1615alimi 1416 . . . . . . . . . 10  |-  ( A. x ( Fun  G  /\  ( G `  x
)  e.  _V )  ->  A. x Fun  G
)
1714, 16syl 14 . . . . . . . . 9  |-  ( ph  ->  A. x Fun  G
)
181719.21bi 1522 . . . . . . . 8  |-  ( ph  ->  Fun  G )
1918adantr 274 . . . . . . 7  |-  ( (
ph  /\  z  e.  On )  ->  Fun  G
)
20 ordon 4372 . . . . . . . 8  |-  Ord  On
2120a1i 9 . . . . . . 7  |-  ( (
ph  /\  z  e.  On )  ->  Ord  On )
22 simpr 109 . . . . . . . . . . 11  |-  ( ( Fun  G  /\  ( G `  x )  e.  _V )  ->  ( G `  x )  e.  _V )
2322alimi 1416 . . . . . . . . . 10  |-  ( A. x ( Fun  G  /\  ( G `  x
)  e.  _V )  ->  A. x ( G `
 x )  e. 
_V )
24 fveq2 5389 . . . . . . . . . . . 12  |-  ( x  =  f  ->  ( G `  x )  =  ( G `  f ) )
2524eleq1d 2186 . . . . . . . . . . 11  |-  ( x  =  f  ->  (
( G `  x
)  e.  _V  <->  ( G `  f )  e.  _V ) )
2625spv 1816 . . . . . . . . . 10  |-  ( A. x ( G `  x )  e.  _V  ->  ( G `  f
)  e.  _V )
2714, 23, 263syl 17 . . . . . . . . 9  |-  ( ph  ->  ( G `  f
)  e.  _V )
2827adantr 274 . . . . . . . 8  |-  ( (
ph  /\  z  e.  On )  ->  ( G `
 f )  e. 
_V )
29283ad2ant1 987 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  On )  /\  y  e.  On  /\  f  Fn  y )  ->  ( G `  f )  e.  _V )
30 suceloni 4387 . . . . . . . . 9  |-  ( y  e.  On  ->  suc  y  e.  On )
31 unon 4397 . . . . . . . . 9  |-  U. On  =  On
3230, 31eleq2s 2212 . . . . . . . 8  |-  ( y  e.  U. On  ->  suc  y  e.  On )
3332adantl 275 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  On )  /\  y  e.  U. On )  ->  suc  y  e.  On )
34 suceloni 4387 . . . . . . . 8  |-  ( z  e.  On  ->  suc  z  e.  On )
3534adantl 275 . . . . . . 7  |-  ( (
ph  /\  z  e.  On )  ->  suc  z  e.  On )
362, 19, 21, 29, 33, 35tfr1on 6215 . . . . . 6  |-  ( (
ph  /\  z  e.  On )  ->  suc  z  C_ 
dom  F )
37 vex 2663 . . . . . . 7  |-  z  e. 
_V
3837sucid 4309 . . . . . 6  |-  z  e. 
suc  z
39 ssel2 3062 . . . . . 6  |-  ( ( suc  z  C_  dom  F  /\  z  e.  suc  z )  ->  z  e.  dom  F )
4036, 38, 39sylancl 409 . . . . 5  |-  ( (
ph  /\  z  e.  On )  ->  z  e. 
dom  F )
4140ex 114 . . . 4  |-  ( ph  ->  ( z  e.  On  ->  z  e.  dom  F
) )
4241ssrdv 3073 . . 3  |-  ( ph  ->  On  C_  dom  F )
4313, 42eqssd 3084 . 2  |-  ( ph  ->  dom  F  =  On )
44 df-fn 5096 . 2  |-  ( F  Fn  On  <->  ( Fun  F  /\  dom  F  =  On ) )
455, 43, 44sylanbrc 413 1  |-  ( ph  ->  F  Fn  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1314    = wceq 1316    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394   _Vcvv 2660    C_ wss 3041   U.cuni 3706   Ord word 4254   Oncon0 4255   suc csuc 4257   dom cdm 4509    |` cres 4511   Fun wfun 5087    Fn wfn 5088   ` cfv 5093  recscrecs 6169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-recs 6170
This theorem is referenced by: (None)
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