Theorem fnoei 6452

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 6422	. 2 ⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
2		vex 2740	. . 3 ⊢ 𝑦 ∈ V
3		1on 6423	. . . . 5 ⊢ 1o ∈ On
4	3	elexi 2749	. . . 4 ⊢ 1o ∈ V
5		vex 2740	. . . . . 6 ⊢ 𝑧 ∈ V
6		vex 2740	. . . . . 6 ⊢ 𝑥 ∈ V
7		omexg 6451	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝑥 ∈ V) → (𝑧 ·o 𝑥) ∈ V)
8	5, 6, 7	mp2an 426	. . . . 5 ⊢ (𝑧 ·o 𝑥) ∈ V
9		eqid 2177	. . . . 5 ⊢ (𝑧 ∈ V ↦ (𝑧 ·o 𝑥)) = (𝑧 ∈ V ↦ (𝑧 ·o 𝑥))
10	8, 9	fnmpti 5344	. . . 4 ⊢ (𝑧 ∈ V ↦ (𝑧 ·o 𝑥)) Fn V
11	4, 10	rdgexg 6389	. . 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 6204	1 ⊢ ↑o Fn (On × On)

Syntax hints:   ∈ wcel 2148  Vcvv 2737   ↦ cmpt 4064  Oncon0 4363   × cxp 4624   Fn wfn 5211  ‘cfv 5216  (class class class)co 5874  reccrdg 6369  1_oc1o 6409   ·_o comu 6414   ↑_o coei 6415

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-oexpi 6422

This theorem is referenced by: (None)