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Theorem oev2 7870
Description: Alternate value of ordinal exponentiation. Compare oev 7861. (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 6914 . . . . . 6 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴o 𝐵) = (∅ ↑o ∅))
2 oe0m0 7867 . . . . . 6 (∅ ↑o ∅) = 1o
31, 2syl6eq 2877 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴o 𝐵) = 1o)
4 fveq2 6433 . . . . . . . 8 (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅))
5 1oex 7834 . . . . . . . . 9 1o ∈ V
65rdg0 7783 . . . . . . . 8 (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅) = 1o
74, 6syl6eq 2877 . . . . . . 7 (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) = 1o)
8 inteq 4700 . . . . . . . 8 (𝐵 = ∅ → 𝐵 = ∅)
9 int0 4711 . . . . . . . 8 ∅ = V
108, 9syl6eq 2877 . . . . . . 7 (𝐵 = ∅ → 𝐵 = V)
117, 10ineq12d 4042 . . . . . 6 (𝐵 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = (1o ∩ V))
12 inv1 4195 . . . . . . 7 (1o ∩ V) = 1o
1312a1i 11 . . . . . 6 (𝐴 = ∅ → (1o ∩ V) = 1o)
1411, 13sylan9eqr 2883 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = 1o)
153, 14eqtr4d 2864 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵))
16 oveq1 6912 . . . . . . 7 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
17 oe0m1 7868 . . . . . . . 8 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
1817biimpa 470 . . . . . . 7 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
1916, 18sylan9eqr 2883 . . . . . 6 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐴o 𝐵) = ∅)
2019an32s 644 . . . . 5 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴o 𝐵) = ∅)
21 int0el 4728 . . . . . . . 8 (∅ ∈ 𝐵 𝐵 = ∅)
2221ineq2d 4041 . . . . . . 7 (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∅))
23 in0 4193 . . . . . . 7 ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ∅) = ∅
2422, 23syl6eq 2877 . . . . . 6 (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = ∅)
2524adantl 475 . . . . 5 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵) = ∅)
2620, 25eqtr4d 2864 . . . 4 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵))
2715, 26oe0lem 7860 . . 3 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ 𝐵))
28 inteq 4700 . . . . . . . . . 10 (𝐴 = ∅ → 𝐴 = ∅)
2928, 9syl6eq 2877 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = V)
3029difeq2d 3955 . . . . . . . 8 (𝐴 = ∅ → (V ∖ 𝐴) = (V ∖ V))
31 difid 4178 . . . . . . . 8 (V ∖ V) = ∅
3230, 31syl6eq 2877 . . . . . . 7 (𝐴 = ∅ → (V ∖ 𝐴) = ∅)
3332uneq2d 3994 . . . . . 6 (𝐴 = ∅ → ( 𝐵 ∪ (V ∖ 𝐴)) = ( 𝐵 ∪ ∅))
34 uncom 3984 . . . . . 6 ( 𝐵 ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ 𝐵)
35 un0 4192 . . . . . 6 ( 𝐵 ∪ ∅) = 𝐵
3633, 34, 353eqtr3g 2884 . . . . 5 (𝐴 = ∅ → ((V ∖ 𝐴) ∪ 𝐵) = 𝐵)
3736adantl 475 . . . 4 ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((V ∖ 𝐴) ∪ 𝐵) = 𝐵)
3837ineq2d 4041 . . 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 2864 . 2 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
40 oevn0 7862 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
41 int0el 4728 . . . . . . . . . 10 (∅ ∈ 𝐴 𝐴 = ∅)
4241difeq2d 3955 . . . . . . . . 9 (∅ ∈ 𝐴 → (V ∖ 𝐴) = (V ∖ ∅))
43 dif0 4180 . . . . . . . . 9 (V ∖ ∅) = V
4442, 43syl6eq 2877 . . . . . . . 8 (∅ ∈ 𝐴 → (V ∖ 𝐴) = V)
4544uneq2d 3994 . . . . . . 7 (∅ ∈ 𝐴 → ( 𝐵 ∪ (V ∖ 𝐴)) = ( 𝐵 ∪ V))
46 unv 4196 . . . . . . 7 ( 𝐵 ∪ V) = V
4745, 34, 463eqtr3g 2884 . . . . . 6 (∅ ∈ 𝐴 → ((V ∖ 𝐴) ∪ 𝐵) = V)
4847adantl 475 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((V ∖ 𝐴) ∪ 𝐵) = V)
4948ineq2d 4041 . . . 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 4195 . . . 4 ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ V) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)
5149, 50syl6req 2878 . . 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 2861 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
5339, 52oe0lem 7860 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 386   = wceq 1658  wcel 2166  Vcvv 3414  cdif 3795  cun 3796  cin 3797  c0 4144   cint 4697  cmpt 4952  Oncon0 5963  cfv 6123  (class class class)co 6905  reccrdg 7771  1oc1o 7819   ·o comu 7824  o coe 7825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oexp 7832
This theorem is referenced by: (None)
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