Theorem wfrdmss 8261

Description: The domain of the well-ordered recursion generator is a subclass of 𝐴. 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
wfrdmss	⊢ dom 𝐹 ⊆ 𝐴

Proof of Theorem wfrdmss

Step	Hyp	Ref	Expression
1		wfrrel.1	. . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
2		df-wrecs 8252	. . 3 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
3	1, 2	eqtri 2752	. 2 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
4	3	frrdmss 8247	1 ⊢ dom 𝐹 ⊆ 𝐴

Syntax hints:   = wceq 1540   ⊆ wss 3905  dom cdm 5623   ∘ ccom 5627  2^nd c2nd 7930  frecscfrecs 8220  wrecscwrecs 8251

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494  df-ov 7356  df-frecs 8221  df-wrecs 8252

This theorem is referenced by: (None)