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Theorem oev 7539
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 2625 . . 3 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
2 oveq2 6612 . . . . . 6 (𝑦 = 𝐴 → (𝑥 ·𝑜 𝑦) = (𝑥 ·𝑜 𝐴))
32mpteq2dv 4705 . . . . 5 (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)))
4 rdgeq1 7452 . . . . 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 6150 . . 3 (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧))
71, 6ifbieq2d 4083 . 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 3700 . . 3 (𝑧 = 𝐵 → (1𝑜𝑧) = (1𝑜𝐵))
9 fveq2 6148 . . 3 (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
108, 9ifeq12d 4078 . 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 7511 . 2 𝑜 = (𝑦 ∈ On, 𝑧 ∈ On ↦ if(𝑦 = ∅, (1𝑜𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜)‘𝑧)))
12 1on 7512 . . . . 5 1𝑜 ∈ On
1312elexi 3199 . . . 4 1𝑜 ∈ V
14 difss 3715 . . . 4 (1𝑜𝐵) ⊆ 1𝑜
1513, 14ssexi 4763 . . 3 (1𝑜𝐵) ∈ V
16 fvex 6158 . . 3 (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ V
1715, 16ifex 4128 . 2 if(𝐴 = ∅, (1𝑜𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)) ∈ V
187, 10, 11, 17ovmpt2 6749 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 384   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  c0 3891  ifcif 4058  cmpt 4673  Oncon0 5682  cfv 5847  (class class class)co 6604  reccrdg 7450  1𝑜c1o 7498   ·𝑜 comu 7503  𝑜 coe 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oexp 7511
This theorem is referenced by:  oevn0  7540  oe0m  7543
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