Theorem oav 7796

Description: Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
oav	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oav

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

Step	Hyp	Ref	Expression
1		rdgeq2 7712	. . 3 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦) = rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴))
2	1	fveq1d 6377	. 2 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦)‘𝑧) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝑧))
3		fveq2 6375	. 2 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝑧) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
4		df-oadd 7768	. 2 ⊢ +𝑜 = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦)‘𝑧))
5		fvex 6388	. 2 ⊢ (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵) ∈ V
6	2, 3, 4, 5	ovmpt2 6994	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1652   ∈ wcel 2155  Vcvv 3350   ↦ cmpt 4888  Oncon0 5908  suc csuc 5910  ‘cfv 6068  (class class class)co 6842  reccrdg 7709   +_𝑜 coa 7761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-oadd 7768

This theorem is referenced by:  oa0  7801  oasuc  7809  onasuc  7813  oalim  7817