Theorem fnoei 6619

Description: Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)

Assertion

Ref	Expression
fnoei	⊢ ↑o Fn (On × On)

Proof of Theorem fnoei

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

Step	Hyp	Ref	Expression
1		df-oexpi 6587	. 2 ⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
2		vex 2805	. . 3 ⊢ 𝑦 ∈ V
3		1on 6588	. . . . 5 ⊢ 1o ∈ On
4	3	elexi 2815	. . . 4 ⊢ 1o ∈ V
5		vex 2805	. . . . . 6 ⊢ 𝑧 ∈ V
6		vex 2805	. . . . . 6 ⊢ 𝑥 ∈ V
7		omexg 6618	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝑥 ∈ V) → (𝑧 ·o 𝑥) ∈ V)
8	5, 6, 7	mp2an 426	. . . . 5 ⊢ (𝑧 ·o 𝑥) ∈ V
9		eqid 2231	. . . . 5 ⊢ (𝑧 ∈ V ↦ (𝑧 ·o 𝑥)) = (𝑧 ∈ V ↦ (𝑧 ·o 𝑥))
10	8, 9	fnmpti 5461	. . . 4 ⊢ (𝑧 ∈ V ↦ (𝑧 ·o 𝑥)) Fn V
11	4, 10	rdgexg 6554	. . 3 ⊢ (𝑦 ∈ V → (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦) ∈ V)
12	2, 11	ax-mp 5	. 2 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦) ∈ V
13	1, 12	fnmpoi 6367	1 ⊢ ↑o Fn (On × On)

Syntax hints:   ∈ wcel 2202  Vcvv 2802   ↦ cmpt 4150  Oncon0 4460   × cxp 4723   Fn wfn 5321  ‘cfv 5326  (class class class)co 6017  reccrdg 6534  1_oc1o 6574   ·_o comu 6579   ↑_o coei 6580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-oadd 6585  df-omul 6586  df-oexpi 6587

This theorem is referenced by: (None)