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Theorem fvmptiunrelexplb0da 40373
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 6098 . . . 4 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
31, 2syl 17 . . 3 (𝜑 𝑅 = (dom 𝑅 ∪ ran 𝑅))
43reseq2d 5822 . 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 40372 . 2 (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))
104, 9eqsstrd 3956 1 (𝜑 → ( I ↾ 𝑅) ⊆ (𝐶𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  Vcvv 3444  cun 3882  wss 3884   cuni 4803   ciun 4884  cmpt 5113   I cid 5427  dom cdm 5523  ran crn 5524  cres 5525  Rel wrel 5528  cfv 6328  (class class class)co 7139  0cc0 10530  𝑟crelexp 14374
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-mulcl 10592  ax-i2m1 10598
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-n0 11890  df-relexp 14375
This theorem is referenced by:  fvrcllb0da  40382  fvrtrcllb0da  40424
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