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Theorem frrdmss 8317
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 2727 . 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 8302 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 394  w3a 1084   = wceq 1533  wex 1773  {cab 2704  wral 3057  wss 3947  dom cdm 5680  cres 5682  Predcpred 6307   Fn wfn 6546  cfv 6551  (class class class)co 7424  frecscfrecs 8290
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 2166  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-iota 6503  df-fun 6553  df-fn 6554  df-fv 6559  df-ov 7427  df-frecs 8291
This theorem is referenced by:  wfrdmss  8355
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