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Theorem fvmptiunrelexplb0d 43697
Description: If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
Hypotheses
Ref Expression
fvmptiunrelexplb0d.c 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))
fvmptiunrelexplb0d.r (𝜑𝑅 ∈ V)
fvmptiunrelexplb0d.n (𝜑𝑁 ∈ V)
fvmptiunrelexplb0d.0 (𝜑 → 0 ∈ 𝑁)
Assertion
Ref Expression
fvmptiunrelexplb0d (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))
Distinct variable groups:   𝑛,𝑟,𝑁   𝑅,𝑛,𝑟
Allowed substitution hints:   𝜑(𝑛,𝑟)   𝐶(𝑛,𝑟)

Proof of Theorem fvmptiunrelexplb0d
StepHypRef Expression
1 fvmptiunrelexplb0d.0 . . 3 (𝜑 → 0 ∈ 𝑁)
2 oveq2 7439 . . . 4 (𝑛 = 0 → (𝑅𝑟𝑛) = (𝑅𝑟0))
32ssiun2s 5048 . . 3 (0 ∈ 𝑁 → (𝑅𝑟0) ⊆ 𝑛𝑁 (𝑅𝑟𝑛))
41, 3syl 17 . 2 (𝜑 → (𝑅𝑟0) ⊆ 𝑛𝑁 (𝑅𝑟𝑛))
5 fvmptiunrelexplb0d.r . . 3 (𝜑𝑅 ∈ V)
6 relexp0g 15061 . . 3 (𝑅 ∈ V → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
75, 6syl 17 . 2 (𝜑 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8 fvmptiunrelexplb0d.c . . . 4 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))
9 oveq1 7438 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
109iuneq2d 5022 . . . 4 (𝑟 = 𝑅 𝑛𝑁 (𝑟𝑟𝑛) = 𝑛𝑁 (𝑅𝑟𝑛))
11 fvmptiunrelexplb0d.n . . . . 5 (𝜑𝑁 ∈ V)
12 ovex 7464 . . . . . 6 (𝑅𝑟𝑛) ∈ V
1312rgenw 3065 . . . . 5 𝑛𝑁 (𝑅𝑟𝑛) ∈ V
14 iunexg 7988 . . . . 5 ((𝑁 ∈ V ∧ ∀𝑛𝑁 (𝑅𝑟𝑛) ∈ V) → 𝑛𝑁 (𝑅𝑟𝑛) ∈ V)
1511, 13, 14sylancl 586 . . . 4 (𝜑 𝑛𝑁 (𝑅𝑟𝑛) ∈ V)
168, 10, 5, 15fvmptd3 7039 . . 3 (𝜑 → (𝐶𝑅) = 𝑛𝑁 (𝑅𝑟𝑛))
1716eqcomd 2743 . 2 (𝜑 𝑛𝑁 (𝑅𝑟𝑛) = (𝐶𝑅))
184, 7, 173sstr3d 4038 1 (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  cun 3949  wss 3951   ciun 4991  cmpt 5225   I cid 5577  dom cdm 5685  ran crn 5686  cres 5687  cfv 6561  (class class class)co 7431  0cc0 11155  𝑟crelexp 15058
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-mulcl 11217  ax-i2m1 11223
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-n0 12527  df-relexp 15059
This theorem is referenced by:  fvmptiunrelexplb0da  43698  fvrcllb0d  43706  fvrtrcllb0d  43748
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