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Theorem oev 8128
Description: Value of ordinal exponentiation. (Contributed by NM, 30-Dec-2004.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
oev ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) = if(𝐴 = ∅, (1o𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oev
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2822 . . 3 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
2 oveq2 7153 . . . . . 6 (𝑦 = 𝐴 → (𝑥 ·o 𝑦) = (𝑥 ·o 𝐴))
32mpteq2dv 5153 . . . . 5 (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 ·o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)))
4 rdgeq1 8036 . . . . 5 ((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) → rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o) = rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o))
53, 4syl 17 . . . 4 (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o) = rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o))
65fveq1d 6665 . . 3 (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧))
71, 6ifbieq2d 4488 . 2 (𝑦 = 𝐴 → if(𝑦 = ∅, (1o𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧)) = if(𝐴 = ∅, (1o𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧)))
8 difeq2 4090 . . 3 (𝑧 = 𝐵 → (1o𝑧) = (1o𝐵))
9 fveq2 6663 . . 3 (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
108, 9ifeq12d 4483 . 2 (𝑧 = 𝐵 → if(𝐴 = ∅, (1o𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧)) = if(𝐴 = ∅, (1o𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
11 df-oexp 8097 . 2 o = (𝑦 ∈ On, 𝑧 ∈ On ↦ if(𝑦 = ∅, (1o𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧)))
12 1oex 8099 . . . 4 1o ∈ V
1312difexi 5223 . . 3 (1o𝐵) ∈ V
14 fvex 6676 . . 3 (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ V
1513, 14ifex 4511 . 2 if(𝐴 = ∅, (1o𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)) ∈ V
167, 10, 11, 15ovmpo 7299 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) = if(𝐴 = ∅, (1o𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cdif 3930  c0 4288  ifcif 4463  cmpt 5137  Oncon0 6184  cfv 6348  (class class class)co 7145  reccrdg 8034  1oc1o 8084   ·o comu 8089  o coe 8090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oexp 8097
This theorem is referenced by:  oevn0  8129  oe0m  8132
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