Theorem fvmptiunrelexplb0da 44266

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 6264	. . . 4 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
4	3	reseq2d 5967	. 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 44265	. 2 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅))
10	4, 9	eqsstrd 3972	1 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝐶‘𝑅))

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144  Vcvv 3456   ∪ cun 3904   ⊆ wss 3906  ∪ cuni 4867  ∪ ciun 4951   ↦ cmpt 5183   I cid 5543  dom cdm 5649  ran crn 5650   ↾ cres 5651  Rel wrel 5654  ‘cfv 6523  (class class class)co 7398  0cc0 11075  ↑_𝑟crelexp 15034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-mulcl 11137  ax-i2m1 11143

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-iota 6479  df-fun 6525  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-n0 12484  df-relexp 15035

This theorem is referenced by:  fvrcllb0da  44275  fvrtrcllb0da  44317