Theorem fvmptiunrelexplb0d 40022

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 7158	. . . 4 ⊢ (𝑛 = 0 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟0))
3	2	ssiun2s 4964	. . 3 ⊢ (0 ∈ 𝑁 → (𝑅↑𝑟0) ⊆ ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛))
4	1, 3	syl 17	. 2 ⊢ (𝜑 → (𝑅↑𝑟0) ⊆ ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛))
5		fvmptiunrelexplb0d.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
6		relexp0g 14375	. . 3 ⊢ (𝑅 ∈ V → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7	5, 6	syl 17	. 2 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8		fvmptiunrelexplb0d.c	. . . 4 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))
9		oveq1 7157	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
10	9	iuneq2d 4940	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛))
11		fvmptiunrelexplb0d.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ V)
12		ovex 7183	. . . . . 6 ⊢ (𝑅↑𝑟𝑛) ∈ V
13	12	rgenw 3150	. . . . 5 ⊢ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V
14		iunexg 7658	. . . . 5 ⊢ ((𝑁 ∈ V ∧ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V)
15	11, 13, 14	sylancl 588	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V)
16	8, 10, 5, 15	fvmptd3 6785	. . 3 ⊢ (𝜑 → (𝐶‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛))
17	16	eqcomd 2827	. 2 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) = (𝐶‘𝑅))
18	4, 7, 17	3sstr3d 4012	1 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494   ∪ cun 3933   ⊆ wss 3935  ∪ ciun 4911   ↦ cmpt 5138   I cid 5453  dom cdm 5549  ran crn 5550   ↾ cres 5551  ‘cfv 6349  (class class class)co 7150  0cc0 10531  ↑_𝑟crelexp 14373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-mulcl 10593  ax-i2m1 10599

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-n0 11892  df-relexp 14374

This theorem is referenced by:  fvmptiunrelexplb0da  40023  fvrcllb0d  40031  fvrtrcllb0d  40073