Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvmptiunrelexplb0da Structured version   Visualization version   GIF version

Theorem fvmptiunrelexplb0da 43703
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 6294 . . . 4 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
31, 2syl 17 . . 3 (𝜑 𝑅 = (dom 𝑅 ∪ ran 𝑅))
43reseq2d 5996 . 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 43702 . 2 (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))
104, 9eqsstrd 4017 1 (𝜑 → ( I ↾ 𝑅) ⊆ (𝐶𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  Vcvv 3479  cun 3948  wss 3950   cuni 4906   ciun 4990  cmpt 5224   I cid 5576  dom cdm 5684  ran crn 5685  cres 5686  Rel wrel 5689  cfv 6560  (class class class)co 7432  0cc0 11156  𝑟crelexp 15059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-mulcl 11218  ax-i2m1 11224
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-n0 12529  df-relexp 15060
This theorem is referenced by:  fvrcllb0da  43712  fvrtrcllb0da  43754
  Copyright terms: Public domain W3C validator