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Theorem fvmptiunrelexplb0d 40316
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 7148 . . . 4 (𝑛 = 0 → (𝑅𝑟𝑛) = (𝑅𝑟0))
32ssiun2s 4947 . . 3 (0 ∈ 𝑁 → (𝑅𝑟0) ⊆ 𝑛𝑁 (𝑅𝑟𝑛))
41, 3syl 17 . 2 (𝜑 → (𝑅𝑟0) ⊆ 𝑛𝑁 (𝑅𝑟𝑛))
5 fvmptiunrelexplb0d.r . . 3 (𝜑𝑅 ∈ V)
6 relexp0g 14372 . . 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 7147 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
109iuneq2d 4923 . . . 4 (𝑟 = 𝑅 𝑛𝑁 (𝑟𝑟𝑛) = 𝑛𝑁 (𝑅𝑟𝑛))
11 fvmptiunrelexplb0d.n . . . . 5 (𝜑𝑁 ∈ V)
12 ovex 7173 . . . . . 6 (𝑅𝑟𝑛) ∈ V
1312rgenw 3142 . . . . 5 𝑛𝑁 (𝑅𝑟𝑛) ∈ V
14 iunexg 7650 . . . . 5 ((𝑁 ∈ V ∧ ∀𝑛𝑁 (𝑅𝑟𝑛) ∈ V) → 𝑛𝑁 (𝑅𝑟𝑛) ∈ V)
1511, 13, 14sylancl 589 . . . 4 (𝜑 𝑛𝑁 (𝑅𝑟𝑛) ∈ V)
168, 10, 5, 15fvmptd3 6773 . . 3 (𝜑 → (𝐶𝑅) = 𝑛𝑁 (𝑅𝑟𝑛))
1716eqcomd 2828 . 2 (𝜑 𝑛𝑁 (𝑅𝑟𝑛) = (𝐶𝑅))
184, 7, 173sstr3d 3988 1 (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  wral 3130  Vcvv 3469  cun 3906  wss 3908   ciun 4894  cmpt 5122   I cid 5436  dom cdm 5532  ran crn 5533  cres 5534  cfv 6334  (class class class)co 7140  0cc0 10526  𝑟crelexp 14370
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-mulcl 10588  ax-i2m1 10594
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-n0 11886  df-relexp 14371
This theorem is referenced by:  fvmptiunrelexplb0da  40317  fvrcllb0d  40325  fvrtrcllb0d  40367
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