Theorem fvmptiunrelexplb0da 43674

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 6248	. . . 4 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
4	3	reseq2d 5950	. 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 43673	. 2 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅))
10	4, 9	eqsstrd 3981	1 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝐶‘𝑅))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∪ cun 3912   ⊆ wss 3914  ∪ cuni 4871  ∪ ciun 4955   ↦ cmpt 5188   I cid 5532  dom cdm 5638  ran crn 5639   ↾ cres 5640  Rel wrel 5643  ‘cfv 6511  (class class class)co 7387  0cc0 11068  ↑_𝑟crelexp 14985

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-mulcl 11130  ax-i2m1 11136

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-n0 12443  df-relexp 14986

This theorem is referenced by:  fvrcllb0da  43683  fvrtrcllb0da  43725