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Theorem fvmptiunrelexplb0d 43674
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 5053 . . 3 (0 ∈ 𝑁 → (𝑅𝑟0) ⊆ 𝑛𝑁 (𝑅𝑟𝑛))
41, 3syl 17 . 2 (𝜑 → (𝑅𝑟0) ⊆ 𝑛𝑁 (𝑅𝑟𝑛))
5 fvmptiunrelexplb0d.r . . 3 (𝜑𝑅 ∈ V)
6 relexp0g 15058 . . 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 5027 . . . 4 (𝑟 = 𝑅 𝑛𝑁 (𝑟𝑟𝑛) = 𝑛𝑁 (𝑅𝑟𝑛))
11 fvmptiunrelexplb0d.n . . . . 5 (𝜑𝑁 ∈ V)
12 ovex 7464 . . . . . 6 (𝑅𝑟𝑛) ∈ V
1312rgenw 3063 . . . . 5 𝑛𝑁 (𝑅𝑟𝑛) ∈ V
14 iunexg 7987 . . . . 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 2741 . 2 (𝜑 𝑛𝑁 (𝑅𝑟𝑛) = (𝐶𝑅))
184, 7, 173sstr3d 4042 1 (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cun 3961  wss 3963   ciun 4996  cmpt 5231   I cid 5582  dom cdm 5689  ran crn 5690  cres 5691  cfv 6563  (class class class)co 7431  0cc0 11153  𝑟crelexp 15055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-mulcl 11215  ax-i2m1 11221
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-n0 12525  df-relexp 15056
This theorem is referenced by:  fvmptiunrelexplb0da  43675  fvrcllb0d  43683  fvrtrcllb0d  43725
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