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Theorem oev2 8179
Description: Alternate value of ordinal exponentiation. Compare oev 8170. (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 7179 . . . . . 6 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴o 𝐵) = (∅ ↑o ∅))
2 oe0m0 8176 . . . . . 6 (∅ ↑o ∅) = 1o
31, 2eqtrdi 2789 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴o 𝐵) = 1o)
4 fveq2 6674 . . . . . . . 8 (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅))
5 1oex 8144 . . . . . . . . 9 1o ∈ V
65rdg0 8086 . . . . . . . 8 (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅) = 1o
74, 6eqtrdi 2789 . . . . . . 7 (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = 1o)
8 inteq 4839 . . . . . . . 8 (𝐵 = ∅ → 𝐵 = ∅)
9 int0 4850 . . . . . . . 8 ∅ = V
108, 9eqtrdi 2789 . . . . . . 7 (𝐵 = ∅ → 𝐵 = V)
117, 10ineq12d 4104 . . . . . 6 (𝐵 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = (1o ∩ V))
12 inv1 4283 . . . . . . 7 (1o ∩ V) = 1o
1312a1i 11 . . . . . 6 (𝐴 = ∅ → (1o ∩ V) = 1o)
1411, 13sylan9eqr 2795 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = 1o)
153, 14eqtr4d 2776 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵))
16 oveq1 7177 . . . . . . 7 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
17 oe0m1 8177 . . . . . . . 8 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
1817biimpa 480 . . . . . . 7 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
1916, 18sylan9eqr 2795 . . . . . 6 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐴o 𝐵) = ∅)
2019an32s 652 . . . . 5 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴o 𝐵) = ∅)
21 int0el 4867 . . . . . . . 8 (∅ ∈ 𝐵 𝐵 = ∅)
2221ineq2d 4103 . . . . . . 7 (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∅))
23 in0 4280 . . . . . . 7 ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∅) = ∅
2422, 23eqtrdi 2789 . . . . . 6 (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = ∅)
2524adantl 485 . . . . 5 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = ∅)
2620, 25eqtr4d 2776 . . . 4 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵))
2715, 26oe0lem 8169 . . 3 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵))
28 inteq 4839 . . . . . . . . . 10 (𝐴 = ∅ → 𝐴 = ∅)
2928, 9eqtrdi 2789 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = V)
3029difeq2d 4013 . . . . . . . 8 (𝐴 = ∅ → (V ∖ 𝐴) = (V ∖ V))
31 difid 4259 . . . . . . . 8 (V ∖ V) = ∅
3230, 31eqtrdi 2789 . . . . . . 7 (𝐴 = ∅ → (V ∖ 𝐴) = ∅)
3332uneq2d 4053 . . . . . 6 (𝐴 = ∅ → ( 𝐵 ∪ (V ∖ 𝐴)) = ( 𝐵 ∪ ∅))
34 uncom 4043 . . . . . 6 ( 𝐵 ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ 𝐵)
35 un0 4279 . . . . . 6 ( 𝐵 ∪ ∅) = 𝐵
3633, 34, 353eqtr3g 2796 . . . . 5 (𝐴 = ∅ → ((V ∖ 𝐴) ∪ 𝐵) = 𝐵)
3736adantl 485 . . . 4 ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((V ∖ 𝐴) ∪ 𝐵) = 𝐵)
3837ineq2d 4103 . . 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 2776 . 2 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
40 oevn0 8171 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
41 int0el 4867 . . . . . . . . . 10 (∅ ∈ 𝐴 𝐴 = ∅)
4241difeq2d 4013 . . . . . . . . 9 (∅ ∈ 𝐴 → (V ∖ 𝐴) = (V ∖ ∅))
43 dif0 4261 . . . . . . . . 9 (V ∖ ∅) = V
4442, 43eqtrdi 2789 . . . . . . . 8 (∅ ∈ 𝐴 → (V ∖ 𝐴) = V)
4544uneq2d 4053 . . . . . . 7 (∅ ∈ 𝐴 → ( 𝐵 ∪ (V ∖ 𝐴)) = ( 𝐵 ∪ V))
46 unv 4284 . . . . . . 7 ( 𝐵 ∪ V) = V
4745, 34, 463eqtr3g 2796 . . . . . 6 (∅ ∈ 𝐴 → ((V ∖ 𝐴) ∪ 𝐵) = V)
4847adantl 485 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((V ∖ 𝐴) ∪ 𝐵) = V)
4948ineq2d 4103 . . . 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 4283 . . . 4 ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ V) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)
5149, 50eqtr2di 2790 . . 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 2773 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
5339, 52oe0lem 8169 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 399   = wceq 1542  wcel 2114  Vcvv 3398  cdif 3840  cun 3841  cin 3842  c0 4211   cint 4836  cmpt 5110  Oncon0 6172  cfv 6339  (class class class)co 7170  reccrdg 8074  1oc1o 8124   ·o comu 8129  o coe 8130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-oexp 8137
This theorem is referenced by: (None)
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