Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvmptiunrelexplb0d Structured version   Visualization version   GIF version

Theorem fvmptiunrelexplb0d 38816
 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 6918 . . . 4 (𝑛 = 0 → (𝑅𝑟𝑛) = (𝑅𝑟0))
32ssiun2s 4786 . . 3 (0 ∈ 𝑁 → (𝑅𝑟0) ⊆ 𝑛𝑁 (𝑅𝑟𝑛))
41, 3syl 17 . 2 (𝜑 → (𝑅𝑟0) ⊆ 𝑛𝑁 (𝑅𝑟𝑛))
5 fvmptiunrelexplb0d.r . . 3 (𝜑𝑅 ∈ V)
6 relexp0g 14146 . . 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 6917 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
109iuneq2d 4769 . . . 4 (𝑟 = 𝑅 𝑛𝑁 (𝑟𝑟𝑛) = 𝑛𝑁 (𝑅𝑟𝑛))
11 fvmptiunrelexplb0d.n . . . . 5 (𝜑𝑁 ∈ V)
12 ovex 6942 . . . . . 6 (𝑅𝑟𝑛) ∈ V
1312rgenw 3133 . . . . 5 𝑛𝑁 (𝑅𝑟𝑛) ∈ V
14 iunexg 7409 . . . . 5 ((𝑁 ∈ V ∧ ∀𝑛𝑁 (𝑅𝑟𝑛) ∈ V) → 𝑛𝑁 (𝑅𝑟𝑛) ∈ V)
1511, 13, 14sylancl 580 . . . 4 (𝜑 𝑛𝑁 (𝑅𝑟𝑛) ∈ V)
168, 10, 5, 15fvmptd3 6555 . . 3 (𝜑 → (𝐶𝑅) = 𝑛𝑁 (𝑅𝑟𝑛))
1716eqcomd 2831 . 2 (𝜑 𝑛𝑁 (𝑅𝑟𝑛) = (𝐶𝑅))
184, 7, 173sstr3d 3872 1 (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1656   ∈ wcel 2164  ∀wral 3117  Vcvv 3414   ∪ cun 3796   ⊆ wss 3798  ∪ ciun 4742   ↦ cmpt 4954   I cid 5251  dom cdm 5346  ran crn 5347   ↾ cres 5348  ‘cfv 6127  (class class class)co 6910  0cc0 10259  ↑𝑟crelexp 14144 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-mulcl 10321  ax-i2m1 10327 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-n0 11626  df-relexp 14145 This theorem is referenced by:  fvmptiunrelexplb0da  38817  fvrcllb0d  38825  fvrtrcllb0d  38867
 Copyright terms: Public domain W3C validator