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Theorem oav 5945
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 5935 . . 3 (x V ↦ suc x) Fn V
21rdgexgg 5881 . 2 ((A On B On) → (rec((x V ↦ suc x), A)‘B) V)
3 rdgeq2 5876 . . . 4 (y = A → rec((x V ↦ suc x), y) = rec((x V ↦ suc x), A))
43fveq1d 5101 . . 3 (y = A → (rec((x V ↦ suc x), y)‘z) = (rec((x V ↦ suc x), A)‘z))
5 fveq2 5099 . . 3 (z = B → (rec((x V ↦ suc x), A)‘z) = (rec((x V ↦ suc x), A)‘B))
6 df-oadd 5916 . . 3 +𝑜 = (y On, z On ↦ (rec((x V ↦ suc x), y)‘z))
74, 5, 6ovmpt2g 5554 . 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 1216 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 1226   wcel 1370  Vcvv 2531  cmpt 3788  Oncon0 4045  suc csuc 4047  cfv 4825  (class class class)co 5432  reccrdg 5873   +𝑜 coa 5909
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-id 4000  df-iord 4048  df-on 4050  df-suc 4053  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-recs 5838  df-irdg 5874  df-oadd 5916
This theorem is referenced by:  oa0  5948  oacl  5951  oav2  5954  oawordi  5960
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