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Theorem oev2 8469
Description: Alternate value of ordinal exponentiation. Compare oev 8460. (Contributed by NM, 2-Jan-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oev2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oev2
StepHypRef Expression
1 oveq12 7366 . . . . . 6 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴o 𝐵) = (∅ ↑o ∅))
2 oe0m0 8466 . . . . . 6 (∅ ↑o ∅) = 1o
31, 2eqtrdi 2792 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴o 𝐵) = 1o)
4 fveq2 6842 . . . . . . . 8 (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅))
5 1oex 8422 . . . . . . . . 9 1o ∈ V
65rdg0 8367 . . . . . . . 8 (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅) = 1o
74, 6eqtrdi 2792 . . . . . . 7 (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = 1o)
8 inteq 4910 . . . . . . . 8 (𝐵 = ∅ → 𝐵 = ∅)
9 int0 4923 . . . . . . . 8 ∅ = V
108, 9eqtrdi 2792 . . . . . . 7 (𝐵 = ∅ → 𝐵 = V)
117, 10ineq12d 4173 . . . . . 6 (𝐵 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = (1o ∩ V))
12 inv1 4354 . . . . . . 7 (1o ∩ V) = 1o
1312a1i 11 . . . . . 6 (𝐴 = ∅ → (1o ∩ V) = 1o)
1411, 13sylan9eqr 2798 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = 1o)
153, 14eqtr4d 2779 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵))
16 oveq1 7364 . . . . . . 7 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
17 oe0m1 8467 . . . . . . . 8 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
1817biimpa 477 . . . . . . 7 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
1916, 18sylan9eqr 2798 . . . . . 6 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐴o 𝐵) = ∅)
2019an32s 650 . . . . 5 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴o 𝐵) = ∅)
21 int0el 4940 . . . . . . . 8 (∅ ∈ 𝐵 𝐵 = ∅)
2221ineq2d 4172 . . . . . . 7 (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∅))
23 in0 4351 . . . . . . 7 ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∅) = ∅
2422, 23eqtrdi 2792 . . . . . 6 (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = ∅)
2524adantl 482 . . . . 5 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = ∅)
2620, 25eqtr4d 2779 . . . 4 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵))
2715, 26oe0lem 8459 . . 3 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵))
28 inteq 4910 . . . . . . . . . 10 (𝐴 = ∅ → 𝐴 = ∅)
2928, 9eqtrdi 2792 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = V)
3029difeq2d 4082 . . . . . . . 8 (𝐴 = ∅ → (V ∖ 𝐴) = (V ∖ V))
31 difid 4330 . . . . . . . 8 (V ∖ V) = ∅
3230, 31eqtrdi 2792 . . . . . . 7 (𝐴 = ∅ → (V ∖ 𝐴) = ∅)
3332uneq2d 4123 . . . . . 6 (𝐴 = ∅ → ( 𝐵 ∪ (V ∖ 𝐴)) = ( 𝐵 ∪ ∅))
34 uncom 4113 . . . . . 6 ( 𝐵 ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ 𝐵)
35 un0 4350 . . . . . 6 ( 𝐵 ∪ ∅) = 𝐵
3633, 34, 353eqtr3g 2799 . . . . 5 (𝐴 = ∅ → ((V ∖ 𝐴) ∪ 𝐵) = 𝐵)
3736adantl 482 . . . 4 ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((V ∖ 𝐴) ∪ 𝐵) = 𝐵)
3837ineq2d 4172 . . 3 ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵))
3927, 38eqtr4d 2779 . 2 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
40 oevn0 8461 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
41 int0el 4940 . . . . . . . . . 10 (∅ ∈ 𝐴 𝐴 = ∅)
4241difeq2d 4082 . . . . . . . . 9 (∅ ∈ 𝐴 → (V ∖ 𝐴) = (V ∖ ∅))
43 dif0 4332 . . . . . . . . 9 (V ∖ ∅) = V
4442, 43eqtrdi 2792 . . . . . . . 8 (∅ ∈ 𝐴 → (V ∖ 𝐴) = V)
4544uneq2d 4123 . . . . . . 7 (∅ ∈ 𝐴 → ( 𝐵 ∪ (V ∖ 𝐴)) = ( 𝐵 ∪ V))
46 unv 4355 . . . . . . 7 ( 𝐵 ∪ V) = V
4745, 34, 463eqtr3g 2799 . . . . . 6 (∅ ∈ 𝐴 → ((V ∖ 𝐴) ∪ 𝐵) = V)
4847adantl 482 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((V ∖ 𝐴) ∪ 𝐵) = V)
4948ineq2d 4172 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ V))
50 inv1 4354 . . . 4 ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ V) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)
5149, 50eqtr2di 2793 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
5240, 51eqtrd 2776 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
5339, 52oe0lem 8459 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  cun 3908  cin 3909  c0 4282   cint 4907  cmpt 5188  Oncon0 6317  cfv 6496  (class class class)co 7357  reccrdg 8355  1oc1o 8405   ·o comu 8410  o coe 8411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oexp 8418
This theorem is referenced by: (None)
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