Theorem rdgifnon 5882

Description: The recursive definition generator is a function on ordinal numbers. The 𝐹 Fn V condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough, but being defined for all sets will generally hold for the characteristic functions we need to use this with). (Contributed by Jim Kingdon, 13-Jul-2019.)

Assertion

Ref	Expression
rdgifnon	⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → rec(𝐹, A) Fn On)

Proof of Theorem rdgifnon

Dummy variables f g x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-irdg 5873	. 2 ⊢ rec(𝐹, A) = recs((g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x)))))
2		rdgruledefgg 5877	. . 3 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → (Fun (g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x)))) ∧ ((g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))))‘f) ∈ V))
3	2	alrimiv 1737	. 2 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → ∀f(Fun (g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x)))) ∧ ((g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))))‘f) ∈ V))
4	1, 3	tfri1d 5866	1 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → rec(𝐹, A) Fn On)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1375  Vcvv 2534   ∪ cun 2891  ∪ ciun 3630   ↦ cmpt 3791  Oncon0 4047  dom cdm 4270  Fun wfun 4821   Fn wfn 4822  ‘cfv 4827  reccrdg 5872

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-coll 3845  ax-sep 3848  ax-pow 3900  ax-pr 3917  ax-un 4118  ax-setind 4202

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ne 2189  df-ral 2288  df-rex 2289  df-reu 2290  df-rab 2292  df-v 2536  df-sbc 2741  df-csb 2829  df-dif 2896  df-un 2898  df-in 2900  df-ss 2907  df-nul 3201  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358  df-uni 3554  df-iun 3632  df-br 3738  df-opab 3792  df-mpt 3793  df-tr 3828  df-id 4003  df-iord 4050  df-on 4052  df-suc 4055  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-res 4282  df-ima 4283  df-iota 4792  df-fun 4829  df-fn 4830  df-f 4831  df-f1 4832  df-fo 4833  df-f1o 4834  df-fv 4835  df-recs 5837  df-irdg 5873

This theorem is referenced by:  rdgivallem  5883  frecrdg  5903