Theorem fvmptiunrelexplb0da 44070

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 6243	. . . 4 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
4	3	reseq2d 5948	. 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 44069	. 2 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅))
10	4, 9	eqsstrd 3970	1 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝐶‘𝑅))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903  ∪ cuni 4865  ∪ ciun 4948   ↦ cmpt 5181   I cid 5528  dom cdm 5634  ran crn 5635   ↾ cres 5636  Rel wrel 5639  ‘cfv 6502  (class class class)co 7370  0cc0 11040  ↑_𝑟crelexp 14956

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-mulcl 11102  ax-i2m1 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-iota 6458  df-fun 6504  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-n0 12416  df-relexp 14957

This theorem is referenced by:  fvrcllb0da  44079  fvrtrcllb0da  44121