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Theorem frrdmcl 8295
Description: Show without using the axiom of replacement that for a "function" defined by well-founded recursion, the predecessor class of an element of its domain is a subclass of its domain. (Contributed by Scott Fenton, 21-Apr-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
Hypothesis
Ref Expression
frrrel.1 𝐹 = frecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
frrdmcl (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)

Proof of Theorem frrdmcl
Dummy variables 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 predeq3 6303 . . 3 (𝑧 = 𝑋 → Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, 𝐴, 𝑋))
21sseq1d 4012 . 2 (𝑧 = 𝑋 → (Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹))
3 eqid 2730 . . 3 {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))))}
4 frrrel.1 . . 3 𝐹 = frecs(𝑅, 𝐴, 𝐺)
53, 4frrlem8 8280 . 2 (𝑧 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
62, 5vtoclga 3565 1 (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1085   = wceq 1539  wex 1779  wcel 2104  {cab 2707  wral 3059  wss 3947  dom cdm 5675  cres 5677  Predcpred 6298   Fn wfn 6537  cfv 6542  (class class class)co 7411  frecscfrecs 8267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-ov 7414  df-frecs 8268
This theorem is referenced by:  wfrdmcl  8333
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