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Theorem fvmptiunrelexplb0da 37497
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 5630 . . . 4 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
31, 2syl 17 . . 3 (𝜑 𝑅 = (dom 𝑅 ∪ ran 𝑅))
43reseq2d 5366 . 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 37496 . 2 (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))
104, 9eqsstrd 3624 1 (𝜑 → ( I ↾ 𝑅) ⊆ (𝐶𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3190  cun 3558  wss 3560   cuni 4409   ciun 4492  cmpt 4683   I cid 4994  dom cdm 5084  ran crn 5085  cres 5086  Rel wrel 5089  cfv 5857  (class class class)co 6615  0cc0 9896  𝑟crelexp 13710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-mulcl 9958  ax-i2m1 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-n0 11253  df-relexp 13711
This theorem is referenced by:  fvrcllb0da  37506  fvrtrcllb0da  37548
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