Theorem wfrdmcl 8331

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

Step	Hyp	Ref	Expression
1		wfrrel.1	. . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
2		df-wrecs 8297	. . 3 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
3	1, 2	eqtri 2761	. 2 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
4	3	frrdmcl 8293	1 ⊢ (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107   ⊆ wss 3949  dom cdm 5677   ∘ ccom 5681  Predcpred 6300  2^nd c2nd 7974  frecscfrecs 8265  wrecscwrecs 8296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-ov 7412  df-frecs 8266  df-wrecs 8297

This theorem is referenced by: (None)