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Theorem fvmptiunrelexplb0d 37802
 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 6655 . . . 4 (𝑛 = 0 → (𝑅𝑟𝑛) = (𝑅𝑟0))
32ssiun2s 4562 . . 3 (0 ∈ 𝑁 → (𝑅𝑟0) ⊆ 𝑛𝑁 (𝑅𝑟𝑛))
41, 3syl 17 . 2 (𝜑 → (𝑅𝑟0) ⊆ 𝑛𝑁 (𝑅𝑟𝑛))
5 fvmptiunrelexplb0d.r . . 3 (𝜑𝑅 ∈ V)
6 relexp0g 13756 . . 3 (𝑅 ∈ V → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
75, 6syl 17 . 2 (𝜑 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8 fvmptiunrelexplb0d.n . . . . 5 (𝜑𝑁 ∈ V)
9 ovex 6675 . . . . . 6 (𝑅𝑟𝑛) ∈ V
109rgenw 2923 . . . . 5 𝑛𝑁 (𝑅𝑟𝑛) ∈ V
11 iunexg 7140 . . . . 5 ((𝑁 ∈ V ∧ ∀𝑛𝑁 (𝑅𝑟𝑛) ∈ V) → 𝑛𝑁 (𝑅𝑟𝑛) ∈ V)
128, 10, 11sylancl 694 . . . 4 (𝜑 𝑛𝑁 (𝑅𝑟𝑛) ∈ V)
13 oveq1 6654 . . . . . 6 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
1413iuneq2d 4545 . . . . 5 (𝑟 = 𝑅 𝑛𝑁 (𝑟𝑟𝑛) = 𝑛𝑁 (𝑅𝑟𝑛))
15 fvmptiunrelexplb0d.c . . . . 5 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))
1614, 15fvmptg 6278 . . . 4 ((𝑅 ∈ V ∧ 𝑛𝑁 (𝑅𝑟𝑛) ∈ V) → (𝐶𝑅) = 𝑛𝑁 (𝑅𝑟𝑛))
175, 12, 16syl2anc 693 . . 3 (𝜑 → (𝐶𝑅) = 𝑛𝑁 (𝑅𝑟𝑛))
1817eqcomd 2627 . 2 (𝜑 𝑛𝑁 (𝑅𝑟𝑛) = (𝐶𝑅))
194, 7, 183sstr3d 3645 1 (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1482   ∈ wcel 1989  ∀wral 2911  Vcvv 3198   ∪ cun 3570   ⊆ wss 3572  ∪ ciun 4518   ↦ cmpt 4727   I cid 5021  dom cdm 5112  ran crn 5113   ↾ cres 5114  ‘cfv 5886  (class class class)co 6647  0cc0 9933  ↑𝑟crelexp 13754 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-mulcl 9995  ax-i2m1 10001 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-n0 11290  df-relexp 13755 This theorem is referenced by:  fvmptiunrelexplb0da  37803  fvrcllb0d  37811  fvrtrcllb0d  37853
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