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Theorem oav 5973
 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.
StepHypRef Expression
1 oafnex 5963 . . 3 (x V ↦ suc x) Fn V
21rdgexgg 5905 . 2 ((A On B On) → (rec((x V ↦ suc x), A)‘B) V)
3 rdgeq2 5899 . . . 4 (y = A → rec((x V ↦ suc x), y) = rec((x V ↦ suc x), A))
43fveq1d 5123 . . 3 (y = A → (rec((x V ↦ suc x), y)‘z) = (rec((x V ↦ suc x), A)‘z))
5 fveq2 5121 . . 3 (z = B → (rec((x V ↦ suc x), A)‘z) = (rec((x V ↦ suc x), A)‘B))
6 df-oadd 5944 . . 3 +𝑜 = (y On, z On ↦ (rec((x V ↦ suc x), y)‘z))
74, 5, 6ovmpt2g 5577 . 2 ((A On B On (rec((x V ↦ suc x), A)‘B) V) → (A +𝑜 B) = (rec((x V ↦ suc x), A)‘B))
82, 7mpd3an3 1232 1 ((A On B On) → (A +𝑜 B) = (rec((x V ↦ suc x), A)‘B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ↦ cmpt 3809  Oncon0 4066  suc csuc 4068  ‘cfv 4845  (class class class)co 5455  reccrdg 5896   +𝑜 coa 5937 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-recs 5861  df-irdg 5897  df-oadd 5944 This theorem is referenced by:  oa0  5976  oacl  5979  oav2  5982  oawordi  5988
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