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Theorem tfr2a 5877
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 2037 . . . 4  {  |  On  Fn  `  G `  |`  }  {  |  On  Fn  `  G `  |`  }
21tfrlem9 5876 . . 3  dom recs G recs G `  G ` recs G  |`
3 tfr.1 . . . 4  F recs G
43dmeqi 4479 . . 3  dom  F  dom recs G
52, 4eleq2s 2129 . 2  dom  F recs G `  G ` recs G  |`
63fveq1i 5122 . 2  F `
recs G `
73reseq1i 4551 . . 3  F  |` recs G  |`
87fveq2i 5124 . 2  G `
 F  |`  G ` recs G  |`
95, 6, 83eqtr4g 2094 1  dom  F  F `  G `  F  |`
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390   {cab 2023  wral 2300  wrex 2301   Oncon0 4066   dom cdm 4288    |` cres 4290    Fn wfn 4840   ` cfv 4845  recscrecs 5860
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-iun 3650  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  df-recs 5861
This theorem is referenced by:  tfr0  5878  tfri2d  5891  tfri2  5893
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