Theorem ov2ssiunov2 43713

Description: Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 15096 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)

Hypothesis

Ref	Expression
ov2ssiunov2.def	⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))

Assertion

Ref	Expression
ov2ssiunov2	⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑅 ↑ 𝑀) ⊆ (𝐶‘𝑅))

Distinct variable groups:   𝑛,𝑟,𝐶,𝑁, ↑   𝑛,𝑀   𝑅,𝑟,𝑛   𝑈,𝑛   𝑛,𝑉

Allowed substitution hints:   𝑈(𝑟)   𝑀(𝑟)   𝑉(𝑟)

Proof of Theorem ov2ssiunov2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1139	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → 𝑀 ∈ 𝑁)
2		simpr 484	. . . . . 6 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → 𝑛 = 𝑀)
3	2	oveq2d 7447	. . . . 5 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → (𝑅 ↑ 𝑛) = (𝑅 ↑ 𝑀))
4	3	eleq2d 2827	. . . 4 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → (𝑥 ∈ (𝑅 ↑ 𝑛) ↔ 𝑥 ∈ (𝑅 ↑ 𝑀)))
5	1, 4	rspcedv 3615	. . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑥 ∈ (𝑅 ↑ 𝑀) → ∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛)))
6		ov2ssiunov2.def	. . . . . 6 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))
7	6	eliunov2 43692	. . . . 5 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛)))
8	7	biimprd 248	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛) → 𝑥 ∈ (𝐶‘𝑅)))
9	8	3adant3 1133	. . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛) → 𝑥 ∈ (𝐶‘𝑅)))
10	5, 9	syld 47	. 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑥 ∈ (𝑅 ↑ 𝑀) → 𝑥 ∈ (𝐶‘𝑅)))
11	10	ssrdv 3989	1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑅 ↑ 𝑀) ⊆ (𝐶‘𝑅))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  ∪ ciun 4991   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434

This theorem is referenced by:  dftrcl3  43733  dfrtrcl3  43746