Theorem oeicl 6606

Description: Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)

Assertion

Ref	Expression
oeicl	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)

Proof of Theorem oeicl

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

Step	Hyp	Ref	Expression
1		oeiv 6600	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
2		1on 6567	. . . 4 ⊢ 1o ∈ On
3	2	a1i 9	. . 3 ⊢ (𝐴 ∈ On → 1o ∈ On)
4		vex 2802	. . . . . . 7 ⊢ 𝑦 ∈ V
5		omcl 6605	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ·o 𝐴) ∈ On)
6		oveq1 6007	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ·o 𝐴) = (𝑦 ·o 𝐴))
7		eqid 2229	. . . . . . . 8 ⊢ (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
8	6, 7	fvmptg 5709	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ (𝑦 ·o 𝐴) ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) = (𝑦 ·o 𝐴))
9	4, 5, 8	sylancr 414	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) = (𝑦 ·o 𝐴))
10	9, 5	eqeltrd 2306	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
11	10	ancoms 268	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
12	11	ralrimiva 2603	. . 3 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ On ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
13	3, 12	rdgon 6530	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ On)
14	1, 13	eqeltrd 2306	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ↦ cmpt 4144  Oncon0 4453  ‘cfv 5317  (class class class)co 6000  reccrdg 6513  1_oc1o 6553   ·_o comu 6558   ↑_o coei 6559

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-oadd 6564  df-omul 6565  df-oexpi 6566

This theorem is referenced by: (None)