Theorem fnoe 7830

Description: Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
fnoe	⊢ ↑𝑜 Fn (On × On)

Proof of Theorem fnoe

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oexp 7805	. 2 ⊢ ↑𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1𝑜 ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)))
2		1on 7806	. . . 4 ⊢ 1𝑜 ∈ On
3		difexg 5003	. . . 4 ⊢ (1𝑜 ∈ On → (1𝑜 ∖ 𝑦) ∈ V)
4	2, 3	ax-mp 5	. . 3 ⊢ (1𝑜 ∖ 𝑦) ∈ V
5		fvex 6424	. . 3 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦) ∈ V
6	4, 5	ifex 4325	. 2 ⊢ if(𝑥 = ∅, (1𝑜 ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)) ∈ V
7	1, 6	fnmpt2i 7475	1 ⊢ ↑𝑜 Fn (On × On)

Syntax hints:   = wceq 1653   ∈ wcel 2157  Vcvv 3385   ∖ cdif 3766  ∅c0 4115  ifcif 4277   ↦ cmpt 4922   × cxp 5310  Oncon0 5941   Fn wfn 6096  ‘cfv 6101  (class class class)co 6878  reccrdg 7744  1_𝑜c1o 7792   ·_𝑜 comu 7797   ↑_𝑜 coe 7798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-ord 5944  df-on 5945  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-1o 7799  df-oexp 7805

This theorem is referenced by:  oaabs2  7965  omabs  7967