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Theorem wfrdmcl 8356
Description: The predecessor class of an element of the well-ordered recursion generator's domain is a subset of its domain. 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
wfrdmcl (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)

Proof of Theorem wfrdmcl
StepHypRef Expression
1 wfrrel.1 . . 3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
2 df-wrecs 8322 . . 3 wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
31, 2eqtri 2755 . 2 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
43frrdmcl 8318 1 (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wss 3947  dom cdm 5680  ccom 5684  Predcpred 6307  2nd c2nd 7996  frecscfrecs 8290  wrecscwrecs 8321
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-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-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  df-wrecs 8322
This theorem is referenced by: (None)
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