ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfrlem4 Structured version   Unicode version

Theorem tfrlem4 5870
Description: Lemma for transfinite recursion. is the class of all "acceptable" functions, and  F is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.)
Hypothesis
Ref Expression
tfrlem.1  {  |  On  Fn  `  F `  |`  }
Assertion
Ref Expression
tfrlem4  Fun
Distinct variable groups:   ,,,, F   ,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem4
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . 4  {  |  On  Fn  `  F `  |`  }
21tfrlem3 5867 . . 3  {  |  On  Fn  `  F `  |`  }
32abeq2i 2145 . 2  On  Fn  `  F `  |`
4 fnfun 4939 . . . 4  Fn  Fun
54adantr 261 . . 3  Fn  `  F `  |`  Fun
65rexlimivw 2423 . 2  On  Fn  `  F `  |`  Fun
73, 6sylbi 114 1  Fun
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390   {cab 2023  wral 2300  wrex 2301   Oncon0 4066    |` 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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  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:  tfrlem6  5873
  Copyright terms: Public domain W3C validator