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Theorem oev2 7772
 Description: Alternate value of ordinal exponentiation. Compare oev 7763. (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 6822 . . . . . 6 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝑜 𝐵) = (∅ ↑𝑜 ∅))
2 oe0m0 7769 . . . . . 6 (∅ ↑𝑜 ∅) = 1𝑜
31, 2syl6eq 2810 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝑜 𝐵) = 1𝑜)
4 fveq2 6352 . . . . . . . 8 (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅))
5 1on 7736 . . . . . . . . . 10 1𝑜 ∈ On
65elexi 3353 . . . . . . . . 9 1𝑜 ∈ V
76rdg0 7686 . . . . . . . 8 (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅) = 1𝑜
84, 7syl6eq 2810 . . . . . . 7 (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) = 1𝑜)
9 inteq 4630 . . . . . . . 8 (𝐵 = ∅ → 𝐵 = ∅)
10 int0 4642 . . . . . . . 8 ∅ = V
119, 10syl6eq 2810 . . . . . . 7 (𝐵 = ∅ → 𝐵 = V)
128, 11ineq12d 3958 . . . . . 6 (𝐵 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵) = (1𝑜 ∩ V))
13 inv1 4113 . . . . . . 7 (1𝑜 ∩ V) = 1𝑜
1413a1i 11 . . . . . 6 (𝐴 = ∅ → (1𝑜 ∩ V) = 1𝑜)
1512, 14sylan9eqr 2816 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵) = 1𝑜)
163, 15eqtr4d 2797 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵))
17 oveq1 6820 . . . . . . 7 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
18 oe0m1 7770 . . . . . . . 8 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
1918biimpa 502 . . . . . . 7 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
2017, 19sylan9eqr 2816 . . . . . 6 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐴𝑜 𝐵) = ∅)
2120an32s 881 . . . . 5 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴𝑜 𝐵) = ∅)
22 int0el 4660 . . . . . . . 8 (∅ ∈ 𝐵 𝐵 = ∅)
2322ineq2d 3957 . . . . . . 7 (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∅))
24 in0 4111 . . . . . . 7 ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∅) = ∅
2523, 24syl6eq 2810 . . . . . 6 (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵) = ∅)
2625adantl 473 . . . . 5 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵) = ∅)
2721, 26eqtr4d 2797 . . . 4 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵))
2816, 27oe0lem 7762 . . 3 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵))
29 inteq 4630 . . . . . . . . . 10 (𝐴 = ∅ → 𝐴 = ∅)
3029, 10syl6eq 2810 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = V)
3130difeq2d 3871 . . . . . . . 8 (𝐴 = ∅ → (V ∖ 𝐴) = (V ∖ V))
32 difid 4091 . . . . . . . 8 (V ∖ V) = ∅
3331, 32syl6eq 2810 . . . . . . 7 (𝐴 = ∅ → (V ∖ 𝐴) = ∅)
3433uneq2d 3910 . . . . . 6 (𝐴 = ∅ → ( 𝐵 ∪ (V ∖ 𝐴)) = ( 𝐵 ∪ ∅))
35 uncom 3900 . . . . . 6 ( 𝐵 ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ 𝐵)
36 un0 4110 . . . . . 6 ( 𝐵 ∪ ∅) = 𝐵
3734, 35, 363eqtr3g 2817 . . . . 5 (𝐴 = ∅ → ((V ∖ 𝐴) ∪ 𝐵) = 𝐵)
3837adantl 473 . . . 4 ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((V ∖ 𝐴) ∪ 𝐵) = 𝐵)
3938ineq2d 3957 . . 3 ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵))
4028, 39eqtr4d 2797 . 2 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
41 oevn0 7764 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
42 int0el 4660 . . . . . . . . . 10 (∅ ∈ 𝐴 𝐴 = ∅)
4342difeq2d 3871 . . . . . . . . 9 (∅ ∈ 𝐴 → (V ∖ 𝐴) = (V ∖ ∅))
44 dif0 4093 . . . . . . . . 9 (V ∖ ∅) = V
4543, 44syl6eq 2810 . . . . . . . 8 (∅ ∈ 𝐴 → (V ∖ 𝐴) = V)
4645uneq2d 3910 . . . . . . 7 (∅ ∈ 𝐴 → ( 𝐵 ∪ (V ∖ 𝐴)) = ( 𝐵 ∪ V))
47 unv 4114 . . . . . . 7 ( 𝐵 ∪ V) = V
4846, 35, 473eqtr3g 2817 . . . . . 6 (∅ ∈ 𝐴 → ((V ∖ 𝐴) ∪ 𝐵) = V)
4948adantl 473 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((V ∖ 𝐴) ∪ 𝐵) = V)
5049ineq2d 3957 . . . 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))
51 inv1 4113 . . . 4 ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ V) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)
5250, 51syl6req 2811 . . 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)))
5341, 52eqtrd 2794 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
5440, 53oe0lem 7762 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 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ∖ cdif 3712   ∪ cun 3713   ∩ cin 3714  ∅c0 4058  ∩ cint 4627   ↦ cmpt 4881  Oncon0 5884  ‘cfv 6049  (class class class)co 6813  reccrdg 7674  1𝑜c1o 7722   ·𝑜 comu 7727   ↑𝑜 coe 7728 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oexp 7735 This theorem is referenced by: (None)
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