Theorem rdgifnon 6182

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 6189; in cases like df-oadd 6223 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 6173	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
2		rdgruledefgg 6178	. . 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 1809	. 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 6138	1 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1445  Vcvv 2633   ∪ cun 3011  ∪ ciun 3752   ↦ cmpt 3921  Oncon0 4214  dom cdm 4467  Fun wfun 5043   Fn wfn 5044  ‘cfv 5049  reccrdg 6172

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-recs 6108  df-irdg 6173

This theorem is referenced by:  rdgivallem  6184