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Theorem relexp0a 42457
Description: Absorption 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 12473 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 oveq2 7416 . . . . . . . 8 (𝑥 = 1 → (𝐴𝑟𝑥) = (𝐴𝑟1))
32oveq1d 7423 . . . . . . 7 (𝑥 = 1 → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟1)↑𝑟0))
43sseq1d 4013 . . . . . 6 (𝑥 = 1 → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟1)↑𝑟0) ⊆ (𝐴𝑟0)))
54imbi2d 340 . . . . 5 (𝑥 = 1 → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟1)↑𝑟0) ⊆ (𝐴𝑟0))))
6 oveq2 7416 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑟𝑥) = (𝐴𝑟𝑦))
76oveq1d 7423 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟𝑦)↑𝑟0))
87sseq1d 4013 . . . . . 6 (𝑥 = 𝑦 → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)))
98imbi2d 340 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0))))
10 oveq2 7416 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝐴𝑟𝑥) = (𝐴𝑟(𝑦 + 1)))
1110oveq1d 7423 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟(𝑦 + 1))↑𝑟0))
1211sseq1d 4013 . . . . . 6 (𝑥 = (𝑦 + 1) → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0)))
1312imbi2d 340 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0))))
14 oveq2 7416 . . . . . . . 8 (𝑥 = 𝑁 → (𝐴𝑟𝑥) = (𝐴𝑟𝑁))
1514oveq1d 7423 . . . . . . 7 (𝑥 = 𝑁 → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟𝑁)↑𝑟0))
1615sseq1d 4013 . . . . . 6 (𝑥 = 𝑁 → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
1716imbi2d 340 . . . . 5 (𝑥 = 𝑁 → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))))
18 relexp1g 14972 . . . . . . 7 (𝐴𝑉 → (𝐴𝑟1) = 𝐴)
1918oveq1d 7423 . . . . . 6 (𝐴𝑉 → ((𝐴𝑟1)↑𝑟0) = (𝐴𝑟0))
20 ssid 4004 . . . . . 6 (𝐴𝑟0) ⊆ (𝐴𝑟0)
2119, 20eqsstrdi 4036 . . . . 5 (𝐴𝑉 → ((𝐴𝑟1)↑𝑟0) ⊆ (𝐴𝑟0))
22 simp2 1137 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → 𝐴𝑉)
23 simp1 1136 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → 𝑦 ∈ ℕ)
24 relexpsucnnr 14971 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ ℕ) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
2524oveq1d 7423 . . . . . . . . 9 ((𝐴𝑉𝑦 ∈ ℕ) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) = (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0))
2622, 23, 25syl2anc 584 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) = (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0))
27 ovex 7441 . . . . . . . . . . . . 13 (𝐴𝑟𝑦) ∈ V
28 coexg 7919 . . . . . . . . . . . . 13 (((𝐴𝑟𝑦) ∈ V ∧ 𝐴𝑉) → ((𝐴𝑟𝑦) ∘ 𝐴) ∈ V)
2927, 28mpan 688 . . . . . . . . . . . 12 (𝐴𝑉 → ((𝐴𝑟𝑦) ∘ 𝐴) ∈ V)
30 relexp0g 14968 . . . . . . . . . . . 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 5970 . . . . . . . . . . . . 13 dom ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ dom 𝐴
33 rncoss 5971 . . . . . . . . . . . . 13 ran ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ ran (𝐴𝑟𝑦)
34 unss12 4182 . . . . . . . . . . . . 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 6009 . . . . . . . . . . . 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 4036 . . . . . . . . . 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 5995 . . . . . . . . . . 11 ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))) = (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴𝑟𝑦)))
41 ssun1 4172 . . . . . . . . . . . . . . 15 dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
42 ssres2 6009 . . . . . . . . . . . . . . 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 14968 . . . . . . . . . . . . . 14 (𝐴𝑉 → (𝐴𝑟0) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
4543, 44sseqtrrid 4035 . . . . . . . . . . . . 13 (𝐴𝑉 → ( I ↾ dom 𝐴) ⊆ (𝐴𝑟0))
4645adantr 481 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ dom 𝐴) ⊆ (𝐴𝑟0))
47 ssun2 4173 . . . . . . . . . . . . . . 15 ran (𝐴𝑟𝑦) ⊆ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦))
48 ssres2 6009 . . . . . . . . . . . . . . 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 14968 . . . . . . . . . . . . . . 15 ((𝐴𝑟𝑦) ∈ V → ((𝐴𝑟𝑦)↑𝑟0) = ( I ↾ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦))))
5127, 50ax-mp 5 . . . . . . . . . . . . . 14 ((𝐴𝑟𝑦)↑𝑟0) = ( I ↾ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦)))
5249, 51sseqtrri 4019 . . . . . . . . . . . . 13 ( I ↾ ran (𝐴𝑟𝑦)) ⊆ ((𝐴𝑟𝑦)↑𝑟0)
53 simpr 485 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0))
5452, 53sstrid 3993 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ ran (𝐴𝑟𝑦)) ⊆ (𝐴𝑟0))
5546, 54unssd 4186 . . . . . . . . . . 11 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴𝑟𝑦))) ⊆ (𝐴𝑟0))
5640, 55eqsstrid 4030 . . . . . . . . . 10 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))) ⊆ (𝐴𝑟0))
57563adant1 1130 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))) ⊆ (𝐴𝑟0))
5839, 57sstrd 3992 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0) ⊆ (𝐴𝑟0))
5926, 58eqsstrd 4020 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0))
60593exp 1119 . . . . . 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 12229 . . . 4 (𝑁 ∈ ℕ → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
63 oveq2 7416 . . . . . . . 8 (𝑁 = 0 → (𝐴𝑟𝑁) = (𝐴𝑟0))
6463oveq1d 7423 . . . . . . 7 (𝑁 = 0 → ((𝐴𝑟𝑁)↑𝑟0) = ((𝐴𝑟0)↑𝑟0))
65 relexp0idm 42456 . . . . . . 7 (𝐴𝑉 → ((𝐴𝑟0)↑𝑟0) = (𝐴𝑟0))
6664, 65sylan9eq 2792 . . . . . 6 ((𝑁 = 0 ∧ 𝐴𝑉) → ((𝐴𝑟𝑁)↑𝑟0) = (𝐴𝑟0))
67 eqimss 4040 . . . . . 6 (((𝐴𝑟𝑁)↑𝑟0) = (𝐴𝑟0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
6866, 67syl 17 . . . . 5 ((𝑁 = 0 ∧ 𝐴𝑉) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
6968ex 413 . . . 4 (𝑁 = 0 → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
7062, 69jaoi 855 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
711, 70sylbi 216 . 2 (𝑁 ∈ ℕ0 → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
7271impcom 408 1 ((𝐴𝑉𝑁 ∈ ℕ0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3474  cun 3946  wss 3948   I cid 5573  dom cdm 5676  ran crn 5677  cres 5678  ccom 5680  (class class class)co 7408  0cc0 11109  1c1 11110   + caddc 11112  cn 12211  0cn0 12471  𝑟crelexp 14965
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-n0 12472  df-z 12558  df-uz 12822  df-seq 13966  df-relexp 14966
This theorem is referenced by: (None)
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