MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfr1ALT Structured version   Visualization version   GIF version

Theorem tfr1ALT 7649
Description: Alternate proof of tfr1 7646 using well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
tfrALT.1 𝐹 = recs(𝐺)
Assertion
Ref Expression
tfr1ALT 𝐹 Fn On

Proof of Theorem tfr1ALT
StepHypRef Expression
1 epweon 7130 . 2 E We On
2 epse 5232 . 2 E Se On
3 tfrALT.1 . . 3 𝐹 = recs(𝐺)
4 df-recs 7621 . . 3 recs(𝐺) = wrecs( E , On, 𝐺)
53, 4eqtri 2793 . 2 𝐹 = wrecs( E , On, 𝐺)
61, 2, 5wfr1 7586 1 𝐹 Fn On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631   E cep 5161  Oncon0 5866   Fn wfn 6026  wrecscwrecs 7558  recscrecs 7620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-wrecs 7559  df-recs 7621
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator