Theorem oav2 6488

Description: Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.)

Assertion

Ref	Expression
oav2	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oav2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oafnex 6469	. . 3 ⊢ (𝑦 ∈ V ↦ suc 𝑦) Fn V
2		rdgival 6407	. . 3 ⊢ (((𝑦 ∈ V ↦ suc 𝑦) Fn V ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))))
3	1, 2	mp3an1 1335	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))))
4		oav 6479	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵))
5		onelon 4402	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
6		vex 2755	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
7		oaexg 6473	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ V) → (𝐴 +o 𝑥) ∈ V)
8	6, 7	mpan2 425	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 +o 𝑥) ∈ V)
9		sucexg 4515	. . . . . . . . . 10 ⊢ ((𝐴 +o 𝑥) ∈ V → suc (𝐴 +o 𝑥) ∈ V)
10	8, 9	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ On → suc (𝐴 +o 𝑥) ∈ V)
11		suceq 4420	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 +o 𝑥) → suc 𝑦 = suc (𝐴 +o 𝑥))
12		eqid 2189	. . . . . . . . . 10 ⊢ (𝑦 ∈ V ↦ suc 𝑦) = (𝑦 ∈ V ↦ suc 𝑦)
13	11, 12	fvmptg 5613	. . . . . . . . 9 ⊢ (((𝐴 +o 𝑥) ∈ V ∧ suc (𝐴 +o 𝑥) ∈ V) → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +o 𝑥)) = suc (𝐴 +o 𝑥))
14	8, 10, 13	syl2anc 411	. . . . . . . 8 ⊢ (𝐴 ∈ On → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +o 𝑥)) = suc (𝐴 +o 𝑥))
15	14	adantr 276	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +o 𝑥)) = suc (𝐴 +o 𝑥))
16		oav 6479	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
17	16	fveq2d 5538	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ∈ V ↦ suc 𝑦)‘(𝐴 +o 𝑥)) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
18	15, 17	eqtr3d 2224	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → suc (𝐴 +o 𝑥) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
19	5, 18	sylan2 286	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ 𝐵)) → suc (𝐴 +o 𝑥) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
20	19	anassrs 400	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ 𝐵) → suc (𝐴 +o 𝑥) = ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
21	20	iuneq2dv 3922	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥) = ∪ 𝑥 ∈ 𝐵 ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)))
22	21	uneq2d 3304	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥)) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 ((𝑦 ∈ V ↦ suc 𝑦)‘(rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))))
23	3, 4, 22	3eqtr4d 2232	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160  Vcvv 2752   ∪ cun 3142  ∪ ciun 3901   ↦ cmpt 4079  Oncon0 4381  suc csuc 4383   Fn wfn 5230  ‘cfv 5235  (class class class)co 5896  reccrdg 6394   +_o coa 6438

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-oadd 6445

This theorem is referenced by:  oasuc  6489