Theorem fnoe 8496

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 8460	. 2 ⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
2		1on 8467	. . . 4 ⊢ 1o ∈ On
3		difexg 5290	. . . 4 ⊢ (1o ∈ On → (1o ∖ 𝑦) ∈ V)
4	2, 3	ax-mp 5	. . 3 ⊢ (1o ∖ 𝑦) ∈ V
5		fvex 6886	. . 3 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦) ∈ V
6	4, 5	ifex 4532	. 2 ⊢ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)) ∈ V
7	1, 6	fnmpoi 8064	1 ⊢ ↑o Fn (On × On)

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∖ cdif 3895  ∅c0 4278  ifcif 4481   ↦ cmpt 5185   × cxp 5645  Oncon0 6351   Fn wfn 6522  ‘cfv 6527  (class class class)co 7408  reccrdg 8395  1_oc1o 8447   ·_o comu 8452   ↑_o coe 8453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6354  df-on 6355  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-fv 6535  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-1o 8454  df-oexp 8460

This theorem is used by:  oaabs2  8636  omabs  8638