Theorem oav2 6674

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 6655	. . 3 ⊢ (𝑦 ∈ V ↦ suc 𝑦) Fn V
2		rdgival 6591	. . 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 1361	. 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 6665	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵))
5		onelon 4487	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
6		vex 2806	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
7		oaexg 6659	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ V) → (𝐴 +o 𝑥) ∈ V)
8	6, 7	mpan2 425	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 +o 𝑥) ∈ V)
9		sucexg 4602	. . . . . . . . . 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 4505	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 +o 𝑥) → suc 𝑦 = suc (𝐴 +o 𝑥))
12		eqid 2231	. . . . . . . . . 10 ⊢ (𝑦 ∈ V ↦ suc 𝑦) = (𝑦 ∈ V ↦ suc 𝑦)
13	11, 12	fvmptg 5731	. . . . . . . . 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 6665	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
17	16	fveq2d 5652	. . . . . . 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 2266	. . . . . 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 3996	. . 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 3363	. 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 2274	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  Vcvv 2803   ∪ cun 3199  ∪ ciun 3975   ↦ cmpt 4155  Oncon0 4466  suc csuc 4468   Fn wfn 5328  ‘cfv 5333  (class class class)co 6028  reccrdg 6578   +_o coa 6622

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-oadd 6629

This theorem is referenced by:  oasuc  6675