Theorem fvmptiunrelexplb0d 44337

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 ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))
fvmptiunrelexplb0d.r	⊢ (𝜑 → 𝑅 ∈ V)
fvmptiunrelexplb0d.n	⊢ (𝜑 → 𝑁 ∈ V)
fvmptiunrelexplb0d.0	⊢ (𝜑 → 0 ∈ 𝑁)

Assertion

Ref	Expression
fvmptiunrelexplb0d	⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅))

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

Allowed substitution hints:   𝜑(𝑛,𝑟)   𝐶(𝑛,𝑟)

Proof of Theorem fvmptiunrelexplb0d

Step	Hyp	Ref	Expression
1		fvmptiunrelexplb0d.0	. . 3 ⊢ (𝜑 → 0 ∈ 𝑁)
2		oveq2 7419	. . . 4 ⊢ (𝑛 = 0 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟0))
3	2	ssiun2s 5015	. . 3 ⊢ (0 ∈ 𝑁 → (𝑅↑𝑟0) ⊆ ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛))
4	1, 3	syl 18	. 2 ⊢ (𝜑 → (𝑅↑𝑟0) ⊆ ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛))
5		fvmptiunrelexplb0d.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
6		relexp0g 15059	. . 3 ⊢ (𝑅 ∈ V → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7	5, 6	syl 18	. 2 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8		fvmptiunrelexplb0d.c	. . . 4 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))
9		oveq1 7418	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
10	9	iuneq2d 4989	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛))
11		fvmptiunrelexplb0d.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ V)
12		ovex 7444	. . . . . 6 ⊢ (𝑅↑𝑟𝑛) ∈ V
13	12	rgenw 3089	. . . . 5 ⊢ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V
14		iunexg 7960	. . . . 5 ⊢ ((𝑁 ∈ V ∧ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V)
15	11, 13, 14	sylancl 597	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V)
16	8, 10, 5, 15	fvmptd3 7014	. . 3 ⊢ (𝜑 → (𝐶‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛))
17	16	eqcomd 2775	. 2 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) = (𝐶‘𝑅))
18	4, 7, 17	3sstr3d 3997	1 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3461   ∪ cun 3909   ⊆ wss 3911  ∪ ciun 4958   ↦ cmpt 5194   I cid 5556  dom cdm 5662  ran crn 5663   ↾ cres 5664  ‘cfv 6537  (class class class)co 7411  0cc0 11100  ↑_𝑟crelexp 15056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-mulcl 11162  ax-i2m1 11168

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-sbc 3752  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-n0 12505  df-relexp 15057

This theorem is referenced by:  fvmptiunrelexplb0da  44338  fvrcllb0d  44346  fvrtrcllb0d  44388