Theorem fnoe 8477

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 8443	. 2 ⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
2		1on 8449	. . . 4 ⊢ 1o ∈ On
3		difexg 5287	. . . 4 ⊢ (1o ∈ On → (1o ∖ 𝑦) ∈ V)
4	2, 3	ax-mp 5	. . 3 ⊢ (1o ∖ 𝑦) ∈ V
5		fvex 6874	. . 3 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦) ∈ V
6	4, 5	ifex 4542	. 2 ⊢ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)) ∈ V
7	1, 6	fnmpoi 8052	1 ⊢ ↑o Fn (On × On)

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∖ cdif 3914  ∅c0 4299  ifcif 4491   ↦ cmpt 5191   × cxp 5639  Oncon0 6335   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390  reccrdg 8380  1_oc1o 8430   ·_o comu 8435   ↑_o coe 8436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-1o 8437  df-oexp 8443

This theorem is referenced by:  oaabs2  8616  omabs  8618