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

Theorem frrdmss 8115
Description: Show without using the axiom of replacement that the domain of the well-founded recursion generator is a subclass of 𝐴. (Contributed by Scott Fenton, 18-Nov-2024.)
Hypothesis
Ref Expression
frrrel.1 𝐹 = frecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
frrdmss dom 𝐹𝐴

Proof of Theorem frrdmss
Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2740 . 2 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2 frrrel.1 . 2 𝐹 = frecs(𝑅, 𝐴, 𝐺)
31, 2frrlem7 8100 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3a 1086   = wceq 1542  wex 1786  {cab 2717  wral 3066  wss 3892  dom cdm 5590  cres 5592  Predcpred 6200   Fn wfn 6427  cfv 6432  (class class class)co 7272  frecscfrecs 8088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-iota 6390  df-fun 6434  df-fn 6435  df-fv 6440  df-ov 7275  df-frecs 8089
This theorem is referenced by:  wfrdmss  8153
  Copyright terms: Public domain W3C validator