ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfr2a Structured version   Unicode version

Theorem tfr2a 5854
Description: A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.)
Hypothesis
Ref Expression
tfr.1  F recs G
Assertion
Ref Expression
tfr2a  dom  F  F `  G `  F  |`

Proof of Theorem tfr2a
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2018 . . . 4  {  |  On  Fn  `  G `  |`  }  {  |  On  Fn  `  G `  |`  }
21tfrlem9 5853 . . 3  dom recs G recs G `  G ` recs G  |`
3 tfr.1 . . . 4  F recs G
43dmeqi 4459 . . 3  dom  F  dom recs G
52, 4eleq2s 2110 . 2  dom  F recs G `  G ` recs G  |`
63fveq1i 5100 . 2  F `
recs G `
73reseq1i 4531 . . 3  F  |` recs G  |`
87fveq2i 5102 . 2  G `
 F  |`  G ` recs G  |`
95, 6, 83eqtr4g 2075 1  dom  F  F `  G `  F  |`
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1226   wcel 1370   {cab 2004  wral 2280  wrex 2281   Oncon0 4045   dom cdm 4268    |` cres 4270    Fn wfn 4820   ` cfv 4825  recscrecs 5837
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-setind 4200
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-id 4000  df-iord 4048  df-on 4050  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-res 4280  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833  df-recs 5838
This theorem is referenced by:  tfri2d  5868  tfri2  5870
  Copyright terms: Public domain W3C validator