Theorem wfrrel 8260

Description: The well-ordered recursion generator generates a relation. Avoids the axiom of replacement. (Contributed by Scott Fenton, 8-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.)

Hypothesis

Ref	Expression
wfrrel.1	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
wfrrel	⊢ Rel 𝐹

Proof of Theorem wfrrel

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 2757	. 2 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
4	3	frrrel 8246	1 ⊢ Rel 𝐹

Syntax hints:   = wceq 1541   ∘ ccom 5626  Rel wrel 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 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-iota 6446  df-fun 6492  df-fn 6493  df-fv 6498  df-ov 7359  df-frecs 8221  df-wrecs 8252

This theorem is referenced by: (None)