Theorem rdgifnon 5903

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 if being used in a manner similar to rdgon 5910; in cases like df-oadd 5937 either presumably could work). (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 5894	. 2 ⊢ rec(𝐹, A) = recs((g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x)))))
2		rdgruledefgg 5899	. . 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 1751	. 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 5887	1 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → rec(𝐹, A) Fn On)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  Vcvv 2551   ∪ cun 2909  ∪ ciun 3647   ↦ cmpt 3808  Oncon0 4065  dom cdm 4287  Fun wfun 4838   Fn wfn 4839  ‘cfv 4844  reccrdg 5893

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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3862  ax-sep 3865  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-id 4020  df-iord 4068  df-on 4070  df-suc 4073  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-recs 5858  df-irdg 5894

This theorem is referenced by:  rdgivallem  5905