Theorem oav2 6630

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 6611	. . 3 ⊢ (𝑦 ∈ V ↦ suc 𝑦) Fn V
2		rdgival 6547	. . 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 1360	. 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 6621	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵))
5		onelon 4481	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
6		vex 2805	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
7		oaexg 6615	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ V) → (𝐴 +o 𝑥) ∈ V)
8	6, 7	mpan2 425	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 +o 𝑥) ∈ V)
9		sucexg 4596	. . . . . . . . . 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 4499	. . . . . . . . . 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 5722	. . . . . . . . 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 6621	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))
17	16	fveq2d 5643	. . . . . . 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 3991	. . 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 3361	. 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 1397   ∈ wcel 2202  Vcvv 2802   ∪ cun 3198  ∪ ciun 3970   ↦ cmpt 4150  Oncon0 4460  suc csuc 4462   Fn wfn 5321  ‘cfv 5326  (class class class)co 6017  reccrdg 6534   +_o coa 6578

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-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-oadd 6585

This theorem is referenced by:  oasuc  6631