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Theorem tfrlem5 5871
Description: Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)
Hypothesis
Ref Expression
tfrlem.1  {  |  On  Fn  `  F `  |`  }
Assertion
Ref Expression
tfrlem5  h  h
Distinct variable groups:   ,,,, h,,, F   ,, h
Allowed substitution hints:   (,,,,)

Proof of Theorem tfrlem5
Dummy variables  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . 3  {  |  On  Fn  `  F `  |`  }
2 vex 2554 . . 3 
_V
31, 2tfrlem3a 5866 . 2  On  Fn  a  `  a  F `  |`  a
4 vex 2554 . . 3  h 
_V
51, 4tfrlem3a 5866 . 2  h  On  h  Fn  a  h `  a  F `  h  |`  a
6 reeanv 2473 . . 3  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  On  Fn  a  `  a  F `  |`  a  On  h  Fn  a  h `  a  F `  h  |`  a
7 simp2ll 970 . . . . . . . . . 10  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  Fn
8 simp3l 931 . . . . . . . . . 10  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h
9 fnbr 4944 . . . . . . . . . 10  Fn
107, 8, 9syl2anc 391 . . . . . . . . 9  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h
11 simp2rl 972 . . . . . . . . . 10  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  h  Fn
12 simp3r 932 . . . . . . . . . 10  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  h
13 fnbr 4944 . . . . . . . . . 10  h  Fn  h
1411, 12, 13syl2anc 391 . . . . . . . . 9  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h
15 elin 3120 . . . . . . . . 9  i^i
1610, 14, 15sylanbrc 394 . . . . . . . 8  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  i^i
17 onin 4089 . . . . . . . . . 10  On  On  i^i  On
18173ad2ant1 924 . . . . . . . . 9  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  i^i  On
19 fnfun 4939 . . . . . . . . . . 11  Fn  Fun
207, 19syl 14 . . . . . . . . . 10  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  Fun
21 inss1 3151 . . . . . . . . . . 11  i^i  C_
22 fndm 4941 . . . . . . . . . . . 12  Fn  dom
237, 22syl 14 . . . . . . . . . . 11  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  dom
2421, 23syl5sseqr 2988 . . . . . . . . . 10  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  i^i  C_  dom
2520, 24jca 290 . . . . . . . . 9  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  Fun  i^i  C_  dom
26 fnfun 4939 . . . . . . . . . . 11  h  Fn  Fun  h
2711, 26syl 14 . . . . . . . . . 10  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  Fun  h
28 inss2 3152 . . . . . . . . . . 11  i^i  C_
29 fndm 4941 . . . . . . . . . . . 12  h  Fn  dom  h
3011, 29syl 14 . . . . . . . . . . 11  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  dom  h
3128, 30syl5sseqr 2988 . . . . . . . . . 10  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  i^i  C_  dom  h
3227, 31jca 290 . . . . . . . . 9  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  Fun  h  i^i  C_  dom  h
33 simp2lr 971 . . . . . . . . . 10  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  a  `  a  F `  |`  a
34 ssralv 2998 . . . . . . . . . 10  i^i 
C_  a  `  a  F `  |`  a  a  i^i  `  a  F `  |`  a
3521, 33, 34mpsyl 59 . . . . . . . . 9  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  a  i^i  `  a  F `  |`  a
36 simp2rr 973 . . . . . . . . . 10  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  a  h `  a  F `  h  |`  a
37 ssralv 2998 . . . . . . . . . 10  i^i 
C_  a  h `  a  F `  h  |`  a  a  i^i  h `  a  F `  h  |`  a
3828, 36, 37mpsyl 59 . . . . . . . . 9  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  a  i^i  h `  a  F `  h  |`  a
3918, 25, 32, 35, 38tfrlem1 5864 . . . . . . . 8  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  a  i^i  `  a  h `  a
40 fveq2 5121 . . . . . . . . . 10  a  `  a  `
41 fveq2 5121 . . . . . . . . . 10  a  h `  a  h `
4240, 41eqeq12d 2051 . . . . . . . . 9  a  `  a  h `  a  `  h `
4342rspcv 2646 . . . . . . . 8  i^i  a  i^i  `  a  h `  a  `  h `
4416, 39, 43sylc 56 . . . . . . 7  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  `  h `
45 funbrfv 5155 . . . . . . . 8  Fun  `
4620, 8, 45sylc 56 . . . . . . 7  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  `
47 funbrfv 5155 . . . . . . . 8  Fun  h  h  h `
4827, 12, 47sylc 56 . . . . . . 7  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h  h `
4944, 46, 483eqtr3d 2077 . . . . . 6  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h
50493exp 1102 . . . . 5  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h
5150rexlimdva 2427 . . . 4  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h
5251rexlimiv 2421 . . 3  On  On  Fn  a  `  a  F `  |`  a  h  Fn  a  h `  a  F `  h  |`  a  h
536, 52sylbir 125 . 2  On  Fn  a  `  a  F `  |`  a  On  h  Fn  a  h `  a  F `  h  |`  a  h
543, 5, 53syl2anb 275 1  h  h
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   w3a 884   wceq 1242   wcel 1390   {cab 2023  wral 2300  wrex 2301    i^i cin 2910    C_ wss 2911   class class class wbr 3755   Oncon0 4066   dom cdm 4288    |` cres 4290   Fun wfun 4839    Fn wfn 4840   ` cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  tfrlem7  5874  tfrexlem  5889
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