Theorem rdgifnon 5885

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 5876	. 2 ⊢ rec(𝐹, A) = recs((g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x)))))
2		rdgruledefgg 5880	. . 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 1736	. 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 5869	1 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → rec(𝐹, A) Fn On)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1374  Vcvv 2533   ∪ cun 2890  ∪ ciun 3629   ↦ cmpt 3790  Oncon0 4047  dom cdm 4270  Fun wfun 4821   Fn wfn 4822  ‘cfv 4827  reccrdg 5875

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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3844  ax-sep 3847  ax-pow 3899  ax-pr 3916  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 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2287  df-rex 2288  df-reu 2289  df-rab 2291  df-v 2535  df-sbc 2740  df-csb 2828  df-dif 2895  df-un 2897  df-in 2899  df-ss 2906  df-nul 3200  df-pw 3334  df-sn 3354  df-pr 3355  df-op 3357  df-uni 3553  df-iun 3631  df-br 3737  df-opab 3791  df-mpt 3792  df-tr 3827  df-id 4002  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 5840  df-irdg 5876

This theorem is referenced by:  rdgivallem  5886  frecrdg  5906