Theorem wfrrel 8323

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 8315	. . 3 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
3	1, 2	eqtri 2785	. 2 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
4	3	frrrel 8309	1 ⊢ Rel 𝐹

This proof depends on syntax axioms:   = wceq 1570   ∘ ccom 5663  Rel wrel 5664  2^nd c2nd 7989  frecscfrecs 8283  wrecscwrecs 8314

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-ov 7420  df-frecs 8284  df-wrecs 8315

This theorem is used by:  tfrlem6  8374