Theorem fnoe 8302

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 8273	. 2 ⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
2		1on 8274	. . . 4 ⊢ 1o ∈ On
3		difexg 5246	. . . 4 ⊢ (1o ∈ On → (1o ∖ 𝑦) ∈ V)
4	2, 3	ax-mp 5	. . 3 ⊢ (1o ∖ 𝑦) ∈ V
5		fvex 6769	. . 3 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦) ∈ V
6	4, 5	ifex 4506	. 2 ⊢ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)) ∈ V
7	1, 6	fnmpoi 7883	1 ⊢ ↑o Fn (On × On)

Syntax hints:   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∖ cdif 3880  ∅c0 4253  ifcif 4456   ↦ cmpt 5153   × cxp 5578  Oncon0 6251   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255  reccrdg 8211  1_oc1o 8260   ·_o comu 8265   ↑_o coe 8266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-1o 8267  df-oexp 8273

This theorem is referenced by:  oaabs2  8439  omabs  8441