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Theorem relexp0a 40067
Description: Absorbtion law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.)
Assertion
Ref Expression
relexp0a ((𝐴𝑉𝑁 ∈ ℕ0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))

Proof of Theorem relexp0a
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn0 11902 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 oveq2 7167 . . . . . . . 8 (𝑥 = 1 → (𝐴𝑟𝑥) = (𝐴𝑟1))
32oveq1d 7174 . . . . . . 7 (𝑥 = 1 → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟1)↑𝑟0))
43sseq1d 4001 . . . . . 6 (𝑥 = 1 → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟1)↑𝑟0) ⊆ (𝐴𝑟0)))
54imbi2d 343 . . . . 5 (𝑥 = 1 → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟1)↑𝑟0) ⊆ (𝐴𝑟0))))
6 oveq2 7167 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑟𝑥) = (𝐴𝑟𝑦))
76oveq1d 7174 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟𝑦)↑𝑟0))
87sseq1d 4001 . . . . . 6 (𝑥 = 𝑦 → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)))
98imbi2d 343 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0))))
10 oveq2 7167 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝐴𝑟𝑥) = (𝐴𝑟(𝑦 + 1)))
1110oveq1d 7174 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟(𝑦 + 1))↑𝑟0))
1211sseq1d 4001 . . . . . 6 (𝑥 = (𝑦 + 1) → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0)))
1312imbi2d 343 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0))))
14 oveq2 7167 . . . . . . . 8 (𝑥 = 𝑁 → (𝐴𝑟𝑥) = (𝐴𝑟𝑁))
1514oveq1d 7174 . . . . . . 7 (𝑥 = 𝑁 → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟𝑁)↑𝑟0))
1615sseq1d 4001 . . . . . 6 (𝑥 = 𝑁 → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
1716imbi2d 343 . . . . 5 (𝑥 = 𝑁 → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))))
18 relexp1g 14388 . . . . . . 7 (𝐴𝑉 → (𝐴𝑟1) = 𝐴)
1918oveq1d 7174 . . . . . 6 (𝐴𝑉 → ((𝐴𝑟1)↑𝑟0) = (𝐴𝑟0))
20 ssid 3992 . . . . . 6 (𝐴𝑟0) ⊆ (𝐴𝑟0)
2119, 20eqsstrdi 4024 . . . . 5 (𝐴𝑉 → ((𝐴𝑟1)↑𝑟0) ⊆ (𝐴𝑟0))
22 simp2 1133 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → 𝐴𝑉)
23 simp1 1132 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → 𝑦 ∈ ℕ)
24 relexpsucnnr 14387 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ ℕ) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
2524oveq1d 7174 . . . . . . . . 9 ((𝐴𝑉𝑦 ∈ ℕ) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) = (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0))
2622, 23, 25syl2anc 586 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) = (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0))
27 ovex 7192 . . . . . . . . . . . . 13 (𝐴𝑟𝑦) ∈ V
28 coexg 7637 . . . . . . . . . . . . 13 (((𝐴𝑟𝑦) ∈ V ∧ 𝐴𝑉) → ((𝐴𝑟𝑦) ∘ 𝐴) ∈ V)
2927, 28mpan 688 . . . . . . . . . . . 12 (𝐴𝑉 → ((𝐴𝑟𝑦) ∘ 𝐴) ∈ V)
30 relexp0g 14384 . . . . . . . . . . . 12 (((𝐴𝑟𝑦) ∘ 𝐴) ∈ V → (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0) = ( I ↾ (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴))))
3129, 30syl 17 . . . . . . . . . . 11 (𝐴𝑉 → (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0) = ( I ↾ (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴))))
32 dmcoss 5845 . . . . . . . . . . . . 13 dom ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ dom 𝐴
33 rncoss 5846 . . . . . . . . . . . . 13 ran ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ ran (𝐴𝑟𝑦)
34 unss12 4161 . . . . . . . . . . . . 13 ((dom ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ dom 𝐴 ∧ ran ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ ran (𝐴𝑟𝑦)) → (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴𝑟𝑦)))
3532, 33, 34mp2an 690 . . . . . . . . . . . 12 (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))
36 ssres2 5884 . . . . . . . . . . . 12 ((dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴𝑟𝑦)) → ( I ↾ (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴))) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))))
3735, 36ax-mp 5 . . . . . . . . . . 11 ( I ↾ (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴))) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦)))
3831, 37eqsstrdi 4024 . . . . . . . . . 10 (𝐴𝑉 → (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))))
3922, 38syl 17 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))))
40 resundi 5870 . . . . . . . . . . 11 ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))) = (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴𝑟𝑦)))
41 ssun1 4151 . . . . . . . . . . . . . . 15 dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
42 ssres2 5884 . . . . . . . . . . . . . . 15 (dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) → ( I ↾ dom 𝐴) ⊆ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
4341, 42ax-mp 5 . . . . . . . . . . . . . 14 ( I ↾ dom 𝐴) ⊆ ( I ↾ (dom 𝐴 ∪ ran 𝐴))
44 relexp0g 14384 . . . . . . . . . . . . . 14 (𝐴𝑉 → (𝐴𝑟0) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
4543, 44sseqtrrid 4023 . . . . . . . . . . . . 13 (𝐴𝑉 → ( I ↾ dom 𝐴) ⊆ (𝐴𝑟0))
4645adantr 483 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ dom 𝐴) ⊆ (𝐴𝑟0))
47 ssun2 4152 . . . . . . . . . . . . . . 15 ran (𝐴𝑟𝑦) ⊆ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦))
48 ssres2 5884 . . . . . . . . . . . . . . 15 (ran (𝐴𝑟𝑦) ⊆ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦)) → ( I ↾ ran (𝐴𝑟𝑦)) ⊆ ( I ↾ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦))))
4947, 48ax-mp 5 . . . . . . . . . . . . . 14 ( I ↾ ran (𝐴𝑟𝑦)) ⊆ ( I ↾ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦)))
50 relexp0g 14384 . . . . . . . . . . . . . . 15 ((𝐴𝑟𝑦) ∈ V → ((𝐴𝑟𝑦)↑𝑟0) = ( I ↾ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦))))
5127, 50ax-mp 5 . . . . . . . . . . . . . 14 ((𝐴𝑟𝑦)↑𝑟0) = ( I ↾ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦)))
5249, 51sseqtrri 4007 . . . . . . . . . . . . 13 ( I ↾ ran (𝐴𝑟𝑦)) ⊆ ((𝐴𝑟𝑦)↑𝑟0)
53 simpr 487 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0))
5452, 53sstrid 3981 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ ran (𝐴𝑟𝑦)) ⊆ (𝐴𝑟0))
5546, 54unssd 4165 . . . . . . . . . . 11 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴𝑟𝑦))) ⊆ (𝐴𝑟0))
5640, 55eqsstrid 4018 . . . . . . . . . 10 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))) ⊆ (𝐴𝑟0))
57563adant1 1126 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))) ⊆ (𝐴𝑟0))
5839, 57sstrd 3980 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0) ⊆ (𝐴𝑟0))
5926, 58eqsstrd 4008 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0))
60593exp 1115 . . . . . 6 (𝑦 ∈ ℕ → (𝐴𝑉 → (((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0))))
6160a2d 29 . . . . 5 (𝑦 ∈ ℕ → ((𝐴𝑉 → ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → (𝐴𝑉 → ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0))))
625, 9, 13, 17, 21, 61nnind 11659 . . . 4 (𝑁 ∈ ℕ → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
63 oveq2 7167 . . . . . . . 8 (𝑁 = 0 → (𝐴𝑟𝑁) = (𝐴𝑟0))
6463oveq1d 7174 . . . . . . 7 (𝑁 = 0 → ((𝐴𝑟𝑁)↑𝑟0) = ((𝐴𝑟0)↑𝑟0))
65 relexp0idm 40066 . . . . . . 7 (𝐴𝑉 → ((𝐴𝑟0)↑𝑟0) = (𝐴𝑟0))
6664, 65sylan9eq 2879 . . . . . 6 ((𝑁 = 0 ∧ 𝐴𝑉) → ((𝐴𝑟𝑁)↑𝑟0) = (𝐴𝑟0))
67 eqimss 4026 . . . . . 6 (((𝐴𝑟𝑁)↑𝑟0) = (𝐴𝑟0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
6866, 67syl 17 . . . . 5 ((𝑁 = 0 ∧ 𝐴𝑉) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
6968ex 415 . . . 4 (𝑁 = 0 → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
7062, 69jaoi 853 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
711, 70sylbi 219 . 2 (𝑁 ∈ ℕ0 → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
7271impcom 410 1 ((𝐴𝑉𝑁 ∈ ℕ0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wo 843  w3a 1083   = wceq 1536  wcel 2113  Vcvv 3497  cun 3937  wss 3939   I cid 5462  dom cdm 5558  ran crn 5559  cres 5560  ccom 5562  (class class class)co 7159  0cc0 10540  1c1 10541   + caddc 10543  cn 11641  0cn0 11900  𝑟crelexp 14382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-n0 11901  df-z 11985  df-uz 12247  df-seq 13373  df-relexp 14383
This theorem is referenced by: (None)
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