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

Theorem wfrdmss 8328
Description: The domain of the well-ordered recursion generator is a subclass of 𝐴. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
Hypothesis
Ref Expression
wfrrel.1 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfrdmss dom 𝐹𝐴

Proof of Theorem wfrdmss
StepHypRef Expression
1 wfrrel.1 . . 3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
2 df-wrecs 8295 . . 3 wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
31, 2eqtri 2754 . 2 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
43frrdmss 8290 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wss 3943  dom cdm 5669  ccom 5673  2nd c2nd 7970  frecscfrecs 8263  wrecscwrecs 8294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544  df-ov 7407  df-frecs 8264  df-wrecs 8295
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator