Theorem fvmptiunrelexplb0da 43684

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 ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))
fvmptiunrelexplb0da.r	⊢ (𝜑 → 𝑅 ∈ V)
fvmptiunrelexplb0da.n	⊢ (𝜑 → 𝑁 ∈ V)
fvmptiunrelexplb0da.rel	⊢ (𝜑 → Rel 𝑅)
fvmptiunrelexplb0da.0	⊢ (𝜑 → 0 ∈ 𝑁)

Assertion

Ref	Expression
fvmptiunrelexplb0da	⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝐶‘𝑅))

Distinct variable groups:   𝑛,𝑟,𝐶,𝑁   𝑅,𝑛,𝑟

Allowed substitution hints:   𝜑(𝑛,𝑟)

Proof of Theorem fvmptiunrelexplb0da

Step	Hyp	Ref	Expression
1		fvmptiunrelexplb0da.rel	. . . 4 ⊢ (𝜑 → Rel 𝑅)
2		relfld 6269	. . . 4 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
4	3	reseq2d 5971	. 2 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5		fvmptiunrelexplb0da.c	. . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))
6		fvmptiunrelexplb0da.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
7		fvmptiunrelexplb0da.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ V)
8		fvmptiunrelexplb0da.0	. . 3 ⊢ (𝜑 → 0 ∈ 𝑁)
9	5, 6, 7, 8	fvmptiunrelexplb0d 43683	. 2 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅))
10	4, 9	eqsstrd 3998	1 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝐶‘𝑅))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∪ cun 3929   ⊆ wss 3931  ∪ cuni 4888  ∪ ciun 4972   ↦ cmpt 5206   I cid 5552  dom cdm 5659  ran crn 5660   ↾ cres 5661  Rel wrel 5664  ‘cfv 6536  (class class class)co 7410  0cc0 11134  ↑_𝑟crelexp 15043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-mulcl 11196  ax-i2m1 11202

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-n0 12507  df-relexp 15044

This theorem is referenced by:  fvrcllb0da  43693  fvrtrcllb0da  43735