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Theorem fvmptiunrelexplb0da 43254
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
fvmptiunrelexplb0da.c 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))
fvmptiunrelexplb0da.r (𝜑𝑅 ∈ V)
fvmptiunrelexplb0da.n (𝜑𝑁 ∈ V)
fvmptiunrelexplb0da.rel (𝜑 → Rel 𝑅)
fvmptiunrelexplb0da.0 (𝜑 → 0 ∈ 𝑁)
Assertion
Ref Expression
fvmptiunrelexplb0da (𝜑 → ( I ↾ 𝑅) ⊆ (𝐶𝑅))
Distinct variable groups:   𝑛,𝑟,𝐶,𝑁   𝑅,𝑛,𝑟
Allowed substitution hints:   𝜑(𝑛,𝑟)

Proof of Theorem fvmptiunrelexplb0da
StepHypRef Expression
1 fvmptiunrelexplb0da.rel . . . 4 (𝜑 → Rel 𝑅)
2 relfld 6281 . . . 4 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
31, 2syl 17 . . 3 (𝜑 𝑅 = (dom 𝑅 ∪ ran 𝑅))
43reseq2d 5985 . 2 (𝜑 → ( I ↾ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5 fvmptiunrelexplb0da.c . . 3 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))
6 fvmptiunrelexplb0da.r . . 3 (𝜑𝑅 ∈ V)
7 fvmptiunrelexplb0da.n . . 3 (𝜑𝑁 ∈ V)
8 fvmptiunrelexplb0da.0 . . 3 (𝜑 → 0 ∈ 𝑁)
95, 6, 7, 8fvmptiunrelexplb0d 43253 . 2 (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))
104, 9eqsstrd 4015 1 (𝜑 → ( I ↾ 𝑅) ⊆ (𝐶𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3461  cun 3942  wss 3944   cuni 4909   ciun 4997  cmpt 5232   I cid 5575  dom cdm 5678  ran crn 5679  cres 5680  Rel wrel 5683  cfv 6549  (class class class)co 7419  0cc0 11140  𝑟crelexp 15002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-mulcl 11202  ax-i2m1 11208
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-n0 12506  df-relexp 15003
This theorem is referenced by:  fvrcllb0da  43263  fvrtrcllb0da  43305
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