Theorem wfrdmcl 8265

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 8255	. . 3 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
3	1, 2	eqtri 2760	. 2 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
4	3	frrdmcl 8251	1 ⊢ (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  dom cdm 5624   ∘ ccom 5628  Predcpred 6258  2^nd c2nd 7934  frecscfrecs 8223  wrecscwrecs 8254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-ov 7363  df-frecs 8224  df-wrecs 8255

This theorem is referenced by: (None)