Theorem fnoe 8434

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 8400	. 2 ⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
2		1on 8406	. . . 4 ⊢ 1o ∈ On
3		difexg 5259	. . . 4 ⊢ (1o ∈ On → (1o ∖ 𝑦) ∈ V)
4	2, 3	ax-mp 5	. . 3 ⊢ (1o ∖ 𝑦) ∈ V
5		fvex 6842	. . 3 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦) ∈ V
6	4, 5	ifex 4507	. 2 ⊢ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)) ∈ V
7	1, 6	fnmpoi 8012	1 ⊢ ↑o Fn (On × On)

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3427   ∖ cdif 3882  ∅c0 4263  ifcif 4456   ↦ cmpt 5155   × cxp 5618  Oncon0 6312   Fn wfn 6482  ‘cfv 6487  (class class class)co 7356  reccrdg 8337  1_oc1o 8387   ·_o comu 8392   ↑_o coe 8393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ord 6315  df-on 6316  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fv 6495  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-1o 8394  df-oexp 8400

This theorem is referenced by:  oaabs2  8574  omabs  8576