Theorem fvmptiunrelexplb0d 40036

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 7166	. . . 4 ⊢ (𝑛 = 0 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟0))
3	2	ssiun2s 4974	. . 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 14383	. . 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 7165	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
10	9	iuneq2d 4950	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛))
11		fvmptiunrelexplb0d.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ V)
12		ovex 7191	. . . . . 6 ⊢ (𝑅↑𝑟𝑛) ∈ V
13	12	rgenw 3152	. . . . 5 ⊢ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V
14		iunexg 7666	. . . . 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 6793	. . 3 ⊢ (𝜑 → (𝐶‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛))
17	16	eqcomd 2829	. 2 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) = (𝐶‘𝑅))
18	4, 7, 17	3sstr3d 4015	1 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ∀wral 3140  Vcvv 3496   ∪ cun 3936   ⊆ wss 3938  ∪ ciun 4921   ↦ cmpt 5148   I cid 5461  dom cdm 5557  ran crn 5558   ↾ cres 5559  ‘cfv 6357  (class class class)co 7158  0cc0 10539  ↑_𝑟crelexp 14381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-mulcl 10601  ax-i2m1 10607

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-n0 11901  df-relexp 14382

This theorem is referenced by:  fvmptiunrelexplb0da  40037  fvrcllb0d  40045  fvrtrcllb0d  40087